Convex Cocompact Actions in Real Projective Geometry

CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY

JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Abstract. We study a notion of convex cocompactness for (not necessarily irreducible) discrete subgroups of the projective general linear group acting on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good properties of classical convex cocompact subgroups in rank-one Lie groups. Extending our earlier work [DGK3] from the context of projective orthogonal groups, we show that for word hyperbolic groups preserving a properly convex open set in projective space, the above general notion of convex cocompactness is equivalent to a stronger convex cocompactness condition studied by Crampon–Marquis, and also to the condition that the natural inclusion be a projective Anosov representation. We investigate examples.

Contents 1. Introduction 2 2. Reminders 16 3. Basic facts: actions on convex subsets of P(V ) 19 4. Convex sets with bisaturated boundary 23 5. Duality for convex cocompact actions 30 6. Segments in the full orbital limit set 33 7. Convex cocompactness and no segment implies P1-Anosov 38 8. P1-Anosov implies convex cocompactness 40 9. Smoothing out the nonideal boundary 44 10. Properties of convex cocompact groups 47 p,q−1 11. Convex cocompactness in H 54

J.D. was partially supported by an Alfred P. Sloan Foundation fellowship, and by the National Science Foundation under grant DMS 1510254. F.G. and F.K. were partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11- LABX-0007-01) and the grant DynGeo (ANR-16-CE40-0025-01). F.K. was partially supported by the European Research Council under ERC Starting Grant 715982 (DiGGeS). The authors also acknowledge support from the GEAR Network, funded by the National Science Foundation under grants DMS 1107452, 1107263, and 1107367 (“RNMS: GEomet- ric structures And Representation varieties”). Part of this work was completed while F.K. was in residence at the MSRI in Berkeley, California, for the program Geometric Group Theory (Fall 2016) supported by NSF grant DMS 1440140, and at the INI in Cambridge, UK, for the program Nonpositive curvature group actions and cohomology (Spring 2017) supported by EPSRC grant EP/K032208/1. 1 2 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

12. Examples of groups which are convex cocompact in P(V ) 67 Appendix A. Some open questions 73 References 74

1. Introduction In the classical setting of semisimple Lie groups G of real rank one, a discrete subgroup of G is said to be convex cocompact if it acts cocompactly on some nonempty closed convex subset of the Riemannian symmetric space G/K of G. Such subgroups have been abundantly studied, in particular in the context of Kleinian groups and real hyperbolic geometry, where there is a rich world of examples. They are known to display good geometric and dynamical behavior. On the other hand, in higher-rank semisimple Lie groups G, the condition that a discrete subgroup Γ act cocompactly on some nonempty convex subset of the Riemannian symmetric space G/K turns out to be quite restrictive: Kleiner–Leeb [KL] and Quint [Q] proved, for example, that if G is simple and such a subgroup Γ is Zariski-dense in G, then it is in fact a uniform lattice of G. The notion of an Anosov representation of a word hyperbolic group in a higher-rank semisimple Lie group G, introduced by Labourie [L] and generalized by Guichard–Wienhard [GW3], is a much more flexible notion which has earned a central role in higher Teichmüller–Thurston theory, see e.g. [BIW2, BCLS, KLPc, GGKW, BPS]. Anosov representations are defined, not in terms of convex subsets of the Riemannian symmetric space G/K, but instead in terms of a dynamical condition for the action on a certain flag variety, i.e. on a compact homogeneous space G/P . This dynamical condition guarantees many desirable analogies with convex cocompact subgroups in rank one: see e.g. [L, GW3, KLPa, KLPb, KLPc]. It also allows for the definition of certain interesting geometric structures associated to Anosov representations: see e.g. [GW1, GW3, KLPb, GGKW, CTT]. However, natural convex geometric structures associated to Anosov representations have been lacking in general. Such structures could allow geometric intuition to bear more fully on Anosov representations, making them more accessible through familiar geometric constructions such as convex fundamental domains, and potentially unlocking new sources of examples. While there is a rich supply of examples of Anosov representations into higher-rank Lie groups in the case of surface groups or free groups, it has proven difficult to construct examples for more complicated word hyperbolic groups. n One of the goals of this paper is to show that, when G = PGL(R ) is a projective linear group, there are in many cases natural convex cocompact n geometric structures modeled on P(R ) associated to Anosov representations n into G. In particular, we prove that any Anosov representation into PGL(R ) which preserves a nonempty properly convex open subset of the projective CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 3

n space P(R ) satisﬁes a strong notion of convex cocompactness introduced by Crampon–Marquis [CM]. Conversely, we show that convex cocompact sub- n groups of P(R ) in the sense of [CM] always give rise to Anosov representations, which enables us to give new examples of Anosov representations and study their deformation spaces by constructing these geometric structures directly. In [DGK3] we had previously established this close connection between convex cocompactness in projective space and Anosov representations in the case of irreducible representations valued in a projective orthogonal group PO(p, q). One context where a connection between Anosov representations and convex projective structures has been known for some time is the deformation 3 theory of real projective surfaces, for G = PGL(R ) [Go, CGo]. More generally, it follows from work of Benoist [B3] that if a discrete subgroup Γ of n G = PGL(R ) divides (i.e. acts cocompactly on) a strictly convex open sub- n set of P(R ), then Γ is word hyperbolic and the natural inclusion Γ ,→ G is Anosov. n Benoist [B6] also found examples of discrete subgroups of PGL(R ) which divide properly convex open sets that are not strictly convex; these subgroups are not word hyperbolic. In this paper we study a notion of convex n n cocompactness for discrete subgroups of PGL(R ) acting on P(R ) which simultaneously generalizes Crampon–Marquis’s notion and Benoist’s convex divisible sets [B3, B4, B5, B6]. In particular, we show that this notion is stable under deformation into larger projective general linear groups, after the model of quasi-Fuchsian deformations of Fuchsian groups. This yields n examples of nonhyperbolic irreducible discrete subgroups of PGL(R ) which n are convex cocompact in P(R ) but do not divide a properly convex open n subset of P(R ). We now describe our results in more detail.

1.1. Strong convex cocompactness in P(V ) and Anosov representa- n tions. In the whole paper, we fix an integer n ≥ 2 and set V := R . Recall that an open domain Ω in the projective space P(V ) is said to be properly convex if it is convex and bounded in some affine chart, and strictly convex if in addition its boundary does not contain any nontrivial projective line segment. It is said to have C1 boundary if every point of the boundary of Ω has a unique supporting hyperplane. In [CM], Crampon–Marquis introduced a notion of geometrically finite action of a discrete subgroup Γ of PGL(V ) on a strictly convex open domain 1 of P(V ) with C boundary. If cusps are not allowed, this notion reduces to a natural notion of convex cocompact action on such domains. We will call discrete groups Γ with such actions strongly convex cocompact. Definition 1.1. Let Γ ⊂ PGL(V ) be an infinite discrete subgroup. • Let Ω be a Γ-invariant properly convex open subset of P(V ) whose boundary is strictly convex and C1. The action of Γ on Ω is convex orb cocompact if the convex hull in Ω of the orbital limit set ΛΩ (Γ) of 4 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Γ in Ω is nonempty and has compact quotient by Γ. (For convex cocompact actions on non-strictly convex Ω, see Section 1.4.) • The group Γ is strongly convex cocompact in P(V ) if it admits a convex cocompact action on some nonempty properly convex open 1 subset Ω of P(V ) whose boundary is strictly convex and C . orb Here the orbital limit set ΛΩ (Γ) of Γ in Ω is defined as the set of accumulation points in ∂Ω of the Γ-orbit of some point z ∈ Ω, and the convex orb orb hull of ΛΩ (Γ) in Ω as the intersection of Ω with the convex hull of ΛΩ (Γ) orb in the convex set Ω. It is easy to see that ΛΩ (Γ) does not depend on the choice of z since Ω is strictly convex. We deduce the following remark. Remark 1.2. The action of Γ on the strictly convex open set Ω is convex cocompact in the sense of Definition 1.1 if and only if Γ acts cocompactly on some nonempty closed convex subset of Ω. p,1 p Example 1.3. For V = R with p ≥ 2, any discrete subgroup of Isom(H ) = PO(p, 1) ⊂ PGL(V ) which is convex cocompact in the usual sense is strongly p convex cocompact in P(V ), taking Ω to be the projective model of H . The first main result of this paper is a close connection between strong convex cocompactness in P(V ) and Anosov representations into PGL(V ). Let P1 (resp. Pn−1) be the stabilizer in G = PGL(V ) of a line (resp. hyperplane) n of V = R ; it is a maximal proper parabolic subgroup of G, and G/P1 (resp. ∗ G/Pn−1) identifies with P(V ) (resp. with the dual projective space P(V )). ∗ We shall think of P(V ) as the space of projective hyperplanes in P(V ). Let Γ be a word hyperbolic group. A P1-Anosov representation (sometimes also called a projective Anosov representation) of Γ into G is a representation ρ :Γ → G for which there exist two continuous, ρ-equivariant boundary ∗ ∗ maps ξ : ∂∞Γ → P(V ) and ξ : ∂∞Γ → P(V ) which ∗ (A1) are compatible, i.e. ξ(η) ∈ ξ (η) for all η ∈ ∂∞Γ, ∗ 0 0 (A2) are transverse, i.e. ξ(η) ∈/ ξ (η ) for all η 6= η in ∂∞Γ, (A3) have an associated flow with some uniform contraction/expansion property described in [L, GW3]. We do not state condition (A3) precisely, since we will use in place of it a simple condition on eigenvalues or singular values described in Definition 2.4 and Fact 2.5 below, taken from [GGKW]. Recall that an element g ∈ PGL(V ) is said to be proximal in P(V ) if it admits a unique attracting fixed point in P(V ) (see Section 2.3). A consequence of (A3) is that every infinite-order el- ∗ ement of ρ(Γ) is proximal in P(V ) and in P(V ), and that the image ξ(∂∞Γ) ∗ (resp. ξ (∂∞Γ)) of the boundary map is the proximal limit set of ρ(Γ) in ∗ P(V ) (resp. P(V )), i.e. the closure of the set of attracting fixed points of proximal elements (Definition 2.2). By [GW3, Prop. 4.10], if ρ is irreducible then condition (A3) is automatically satisfied as soon as (A1) and (A2) are, but this is not true in general: see [GGKW, Ex. 7.15]. It is well known (see [GW3, § 6.1 & Th. 4.3]) that a discrete subgroup of PO(p, 1) is convex cocompact in the classical sense if and only if it is CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 5

p+1 word hyperbolic and the natural inclusion Γ ,→ PO(p, 1) ,→ PGL(R ) is P1-Anosov. In this paper, we prove the following higher-rank generalization. Theorem 1.4. Let Γ be any infinite discrete subgroup of G = PGL(V ) preserving a nonempty properly convex open subset of P(V ). Then the following are equivalent: (1) Γ is strongly convex cocompact in P(V ); (2) Γ is word hyperbolic and the natural inclusion Γ ,→ G is P1-Anosov. As mentioned above, for Γ acting cocompactly on a strictly convex open set (which is then a divisible strictly convex set), the implication (1) ⇒ (2) follows from work of Benoist [B3]. Remarks 1.5. (a) The fact that a strongly convex cocompact group is word hyperbolic is due to Crampon–Marquis [CM, Th. 1.8]. (b) In the case where Γ is irreducible (i.e. does not preserve any nontrivial n projective subspace of P(V )) and contained in PO(p, q) ⊂ PGL(R ) for ∗ some p, q ∈ N with p + q = n, Theorem 1.4 was first proved in our earlier work [DGK3, Th. 1.11 & Prop. 1.17 & Prop. 3.7]. In that case we actually gave a more precise version of Theorem 1.4 involving the notion of negative/positive proximal limit set: see Section 1.8 below. Our proof of Theorem 1.4 in the present paper uses many of the ideas of [DGK3]. One main improvement here is the treatment of duality in the general case where there is no nonzero Γ-invariant quadratic form. (c) In independent work, Zimmer [Z] also extends [DGK3] by studying a slightly different notion for actions of discrete subgroups Γ of PGL(V ) on properly convex open subsets Ω of P(V ): by definition [Z, Def. 1.8], a subgroup Γ of Aut(Ω) is regular convex cocompact if it acts cocompactly on some nonempty, Γ-invariant, closed, properly convex subset C of Ω such that every extreme point of C in ∂Ω is a C1 extreme point of Ω. By [Z, Th. 1.21], if Γ is an irreducible discrete subgroup of PGL(V ) which is regular convex cocompact in Aut(Ω) for some Ω ⊂ P(V ), then Γ is word hyperbolic and the natural inclusion Γ ,→ PGL(V ) is P1-Anosov. Conversely, by [Z, Th. 1.10], any irreducible P1-Anosov representation Γ ,→ SL(V ) can be composed with an irreducible representation SL(V ) → SL(V 0), for some larger vector space V 0, so that Γ 0 0 0 becomes regular convex cocompact in Aut(Ω ) for some Ω ⊂ P(V ). (d) In this paper, unlike in [DGK3] or [Z], we do not assume Γ to be irreducible. This makes the notion of Anosov representation slightly more involved (condition (A3) above is not automatic), and also adds some subtleties to the notion of convex cocompactness (see e.g. Remarks 1.22 and 4.6). We note that there exist strongly convex cocompact groups in P(V ) which are not irreducible, and whose Zariski closure is not even reductive (which means that there is an invariant linear subspace W of V which does not admit any invariant complementary subspace): see Proposition 10.9 and Remark 10.10. This contrasts with the case of divisible convex sets described in [Ve]. 6 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Remark 1.6. For n = dim(V ) ≥ 3, there exist P1-Anosov representations ρ :Γ → G = PGL(V ) which do not preserve any nonempty properly convex subset of P(V ): see [DGK3, Ex. 5.2 & 5.3]. However, by [DGK3, Th. 1.7], if ∂∞Γ is connected, then any P1-Anosov representation ρ valued in p+q PO(p, q) ⊂ PGL(R ) preserves a nonempty properly convex open subset p+q p+q of P(R ), hence ρ(Γ) is strongly convex cocompact in P(R ) by Theo- rem 1.4. Extending this, [Z, Th. 1.24] gives suﬃcient group-theoretic conditions on Γ for P1-Anosov representations Γ ,→ PGL(V ) to be regular convex cocompact in Aut(Ω) (in the sense of Remark 1.5.(c)) for some Ω ⊂ P(V ). 1.2. Convex projective structures for Anosov representations. We can apply Theorem 1.4 to show that some well-known families of Anosov representations, such as Hitchin representations in odd dimension, naturally give rise to convex cocompact real projective manifolds. Proposition 1.7. Let Γ be a closed surface group of genus ≥ 2 and ρ :Γ → n PSL(R ) a Hitchin representation. n (1) If n is odd, then ρ(Γ) is strongly convex cocompact in P(R ). n (2) If n is even, then ρ(Γ) is not strongly convex cocompact in P(R ); in fact it does not even preserve any nonempty properly convex subset n of P(R ). For statement (1), see also [Z, Cor. 1.31]. This extends [DGK3, Prop. 1.19], 2k+1 about Hitchin representations valued in SO(k + 1, k) ⊂ PSL(R ). Remarks 1.8. (a) The case n = 3 of Proposition 1.7.(1) is due to Choi– Goldman [Go, CGo], who proved that the Hitchin component of 3 Hom(π1(S), PSL(R )) parametrizes the convex projective structures on S for a given closed hyperbolic surface S. (b) Guichard–Wienhard [GW3, Th. 11.3 & 11.5] associated diﬀerent geo- n metric structures to Hitchin representations into PSL(R ). For even n, n their geometric structures are modeled on P(R ) but can never be convex (see Proposition 1.7.(2)). For odd n, their geometric struc- n n ∗ tures are modeled on the space F1,n−1 ⊂ P(R ) × P((R ) ) of pairs n (`, H) where ` is a line of R and H a hyperplane containing `; these geometric structures lack convexity but live on a compact manifold, unlike the geometric structures of Proposition 1.7.(1) for n > 3. We refer to Proposition 12.1 for a more general statement on convex structures for connected open sets of Anosov representations.

1.3. New examples of Anosov representations. We can also use the implication (1) ⇒ (2) of Theorem 1.4 to obtain new examples of Anosov representations by constructing explicit strongly convex cocompact groups in P(V ). Following this strategy, in [DGK3, § 8] we showed that any word hyperbolic right-angled Coxeter group W admits reﬂection-group representations into some PGL(V ) which are P1-Anosov. Extending this approach, CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 7 in [DGK4] we give an explicit description of the deformation spaces of such representations.

1.4. General convex cocompactness in P(V ). We now discuss gener- alizations of Definition 1.1 where the properly convex open set Ω is not assumed to have any regularity at the boundary. These cover a larger class of groups, not necessarily word hyperbolic. Given Remark 1.2, a naive generalization of Definition 1.1 that immedi- ately comes to mind is the following. Definition 1.9. An infinite discrete subgroup Γ of PGL(V ) is naively convex cocompact in P(V ) if it preserves a properly convex open subset Ω of P(V ) and acts cocompactly on some nonempty closed convex subset C of Ω. However, the class of naively convex cocompact subgroups of PGL(V ) is not stable under small deformations: see Remark 3.10.(b). This is linked to the fact that if Γ and Ω are as in Definition 1.9 with ∂Ω not strictly convex, then the set of accumulation points of a Γ-orbit of Ω may depend on the orbit (see Example 3.7). To address this issue, we introduce a notion of limit set that does not depend on a choice of orbit. Definition 1.10. Let Γ ⊂ PGL(V ) be an infinite discrete subgroup and let Ω be a properly convex open subset of P(V ) invariant under Γ. The full orb orbital limit set ΛΩ (Γ) of Γ in Ω is the union of all accumulation points of all Γ-orbits in Ω. Using this notion, we can introduce another generalization of Definition 1.1 which is slightly stronger and has better properties than Definition 1.9: here is the main definition of the paper. Definition 1.11. Let Γ be an infinite discrete subgroup of PGL(V ). • Let Ω be a nonempty Γ-invariant properly convex open subset of P(V ). The action of Γ on Ω is convex cocompact if the convex hull cor orb CΩ (Γ) of the full orbital limit set ΛΩ (Γ) in Ω is nonempty and has compact quotient by Γ. • Γ is convex cocompact in P(V ) if it admits a convex cocompact action on some nonempty properly convex open subset Ω of P(V ). cor In the setting of Definition 1.11, the set Γ\CΩ (Γ) is a compact convex subset of the projective manifold (or orbifold) Γ\Ω which contains all the topology and which is minimal (Lemma 4.1.(2)); we shall call it the convex cor core of Γ\Ω. By analogy, we shall also call CΩ (Γ) the convex core of Ω for Γ. Example 1.12. If Γ is strongly convex cocompact in P(V ), then it is convex cocompact in P(V ). This is immediate from the definitions since when Ω is orb strictly convex, the full orbital limit set ΛΩ (Γ) coincides with the accumulation set of a single Γ-orbit of Ω. See Theorem 1.15 below for a refinement. 8 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Example 1.13. If Γ divides (i.e. preserves and acts cocompactly on) a cor nonempty properly convex open subset Ω of P(V ), then CΩ (Γ) = Ω and Γ is convex cocompact in P(V ). There are many known examples of divisible convex sets which fail to be strictly convex, yielding convex cocompact groups which are not strongly convex cocompact in P(V ): see Example 3.7 for a baby case. The ﬁrst indecomposable examples for dim(V ) = 4 were constructed by Benoist [B6], and further examples were recently constructed for dim(V ) = 4 in [BDL] and for 5 ≤ dim(V ) ≤ 8 in [CLM]. See Section 12.2.1. In Theorem 1.20 we shall give several characterizations of convex cocompactness in P(V ). In particular, we shall prove that the class of subgroups of PGL(V ) that are convex cocompact in P(V ) is precisely the class of holonomy groups of compact properly convex projective orbifolds with strictly convex boundary; this ensures that this class of groups is stable under small deformations, using [CLT1]. Before stating this and other results, let us make the connection with the context of Theorem 1.4 (strong convex cocompactness).

1.5. Word hyperbolic convex cocompact groups in P(V ). By [B3, Th. 1.1], if a discrete subgroup Γ ⊂ PGL(V ) divides a properly convex open subset Ω ⊂ P(V ), then Ω is strictly convex if and only if Γ is word hyperbolic. A similar statement is not true for convex cocompact actions in general. Example 1.14 (see Proposition 1.7.(1) and Corollary 8.10). Let n ≥ 5 be n odd, Γ a closed surface group of genus ≥ 2 and ρ :Γ → PSL(R ) a Hitchin ∗ ∗ representation, with boundary map ξ : ∂∞Γ → P(V ). Then Γ acts convex cocompactly on n [ ∗ Ω := P(R ) r ξ (η), η∈∂∞Γ n which is a properly convex open subset of P(R ) that is not strictly convex. However, we show that for Γ acting convex cocompactly on Ω, the hyperbolicity of Γ is determined by the convexity behavior of ∂Ω at the full orbital limit set: here is an expanded version of Theorem 1.4. Theorem 1.15. Let Γ be an infinite discrete subgroup of PGL(V ). Then the following are equivalent: (i) Γ is strongly convex cocompact in P(V ) (Definition 1.1); (ii) Γ is convex cocompact in P(V ) (Definition 1.11) and word hyperbolic; (iii) Γ is convex cocompact in P(V ) and for any properly convex open set Ω ⊂ P(V ) on which Γ acts convex cocompactly, the full orbital limit orb set ΛΩ (Γ) does not contain any nontrivial projective line segment; (iv) Γ is convex cocompact in P(V ) and for some nonempty properly convex open set Ω ⊂ P(V ) on which Γ acts convex cocompactly, the cor convex core CΩ (Γ) does not contain a PET; (v) Γ preserves a properly convex open subset Ω of P(V ) and acts cocompactly on some closed convex subset C of Ω with nonempty interior CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 9

such that ∂iC := C r C = C ∩ ∂Ω does not contain any nontrivial projective line segment; (vi) Γ is word hyperbolic, the inclusion Γ ,→ PGL(V ) is P1-Anosov, and Γ preserves some nonempty properly convex open subset of P(V ). In this case, there is equality between the following four sets: orb • the orbital limit set ΛΩ (Γ) of Γ in any Γ-invariant properly convex 1 open subset Ω of P(V ) with strictly convex and C boundary as in Definition 1.1 (condition (i)); orb • the full orbital limit set ΛΩ (Γ) for any properly convex open set Ω on which Γ acts convex cocompactly as in Definition 1.11 (conditions (ii), (iii), (iv)); • the segment-free set ∂iC for any convex subset C on which Γ acts cocompactly in a Γ-invariant properly convex open set Ω (condition (v)); • the image of the boundary map ξ : ∂∞Γ → P(V ) of the Anosov representation Γ ,→ PGL(V ) (condition (vi)), which is also the proximal limit set ΛΓ of Γ in P(V ) (Definition 2.2). In condition (iv), we use the following terminology of Benoist [B6].

Definition 1.16. Let C be a properly convex subset of P(V ).A PET (or properly embedded triangle) in C is a nondegenerate planar triangle whose interior is contained in C, but whose edges and vertices are contained in ∂iC = C r C. It is easy to check (see Lemma 6.1) that if Γ acts convex cocompactly on cor Ω, the existence of a PET in CΩ (Γ) obstructs the hyperbolicity of Γ. 1.6. Properties of convex cocompact groups in P(V ). We show that, even in the case of nonhyperbolic discrete groups, the notion of convex cocompactness in P(V ) (Definition 1.11) still has some of the nice properties enjoyed by Anosov representations and convex cocompact subgroups of rank- one Lie groups. In particular, we prove the following. Theorem 1.17. Let Γ be an infinite discrete subgroup of G = PGL(V ). (A) The group Γ is convex cocompact in P(V ) if and only if it is convex ∗ cocompact in P(V ) (for the dual action). (B) If Γ is convex cocompact in P(V ), then it is finitely generated and quasi-isometrically embedded in G. (C) If Γ is convex cocompact in P(V ), then Γ does not contain any unipotent element. (D) If Γ is convex cocompact in P(V ), then there is a neighborhood U ⊂ Hom(Γ,G) of the natural inclusion such that any ρ ∈ U is injective and discrete with image ρ(Γ) convex cocompact in P(V ). (E) If the semisimplification of Γ (Definition 10.5) is convex cocompact in P(V ), then so is Γ. 0 n0 ± ± 0 (F) Let V = R and let i : SL (V ) ,→ SL (V ⊕V ) be the natural inclusion acting trivially on the second factor. If Γ is convex cocompact in 10 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

0 P(V ), then i(Γ)ˆ is convex cocompact in P(V ⊕ V ), where Γˆ is the lift of Γ to SL±(V ) that preserves a properly convex cone of V lifting Ω (see Remark 3.1).

Remark 1.18. The equivalence (i) ⇔ (ii) of Theorem 1.15 shows that Theo- rem 1.17 still holds if all the occurrences of “convex cocompact” are replaced by “strongly convex cocompact”.

While some of the properties of Theorem 1.17 are proved directly from Deﬁnition 1.11, others will be most naturally established using alternative characterizations of convex cocompactness in P(V ) (Theorem 1.20). We refer to Proposition 10.8 for a more general version of property (F). Properties (F) and (D) give a source for many new examples of convex cocompact groups by starting with known examples in P(V ), including them 0 into P(V ⊕V ), and then deforming. This generalizes the picture of Fuchsian groups in PO(2, 1) being deformed into quasi-Fuchsian groups in PO(3, 1). 4 For example, a group that divides a properly convex open subset of P(R ) whose boundary is not strictly convex may always be deformed nontriv- n ially in a larger projective space P(R ) by bending along torus or Klein bottle subgroups. This gives examples of nonhyperbolic irreducible discrete n n subgroups of PGL(R ) which are convex cocompact in P(R ) but do not n divide any properly convex open subset of P(R ). These groups Γ are quasi- n isometrically embedded in PGL(R ) and structurally stable (i.e. there is a n neighborhood of the natural inclusion in Hom(Γ, PGL(R )) which consists entirely of injective representations, for torsion-free Γ) without being word hyperbolic; compare with Sullivan [Su, Th. A]. We refer to Section 12.2 for more details.

1.7. Holonomy groups of convex projective orbifolds. We now give alternative characterizations of convex cocompact subgroups in P(V ). These characterizations are motivated by a familiar picture in rank one: namely, p if Ω = H is the p-dimensional real hyperbolic space and Γ is a convex p cocompact torsion-free subgroup of PO(p, 1) = Isom(H ), then any uniform cor neighborhood Cu of the convex core CΩ (Γ) has strictly convex boundary and the quotient Γ\Cu is a compact hyperbolic manifold with strictly convex boundary. Let us ﬁx some terminology and notation in order to discuss the appropriate generalization of this picture to real projective geometry.

Deﬁnition 1.19. Let C be a nonempty convex subset of P(V ) (not necessarily open nor closed). • The frontier of C is Fr(C) := C r Int(C). • A supporting hyperplane of C at a point z ∈ Fr(C) is a projective hyperplane H such that z ∈ H ∩ C and the lift of H to V bounds a closed halfspace containing a convex cone lifting C. • The ideal boundary of C is ∂iC := C r C. CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 11

• The nonideal boundary of C is ∂nC := C r Int(C) = Fr(C) r ∂iC. Note that if C is open, then ∂iC = Fr(C) and ∂nC = ∅; in this case, it is common in the literature to denote ∂iC simply by ∂C. • The convex set C has strictly convex nonideal boundary if every point z ∈ ∂nC is an extreme point of C. • The convex set C has C1 nonideal boundary if it has a unique supporting hyperplane at each point z ∈ ∂nC. • The convex set C has bisaturated boundary if for any supporting hyperplane H of C, the set H ∩ C ⊂ Fr(C) is either fully contained in ∂iC or fully contained in ∂nC.

Let C be a properly convex subset of P(V ) on which the discrete subgroup Γ acts properly discontinuously and cocompactly. To simplify this intuitive discussion, assume Γ torsion-free, so that the quotient M = Γ\C is a compact properly convex projective manifold, possibly with boundary. The group Γ is called the holonomy group of M. We show that if the boundary ∂M = Γ\∂nC of M is assumed to have some regularity, then Γ is convex cocompact in P(V ) and, conversely, convex cocompact subgroups in P(V ) are holonomy groups of properly convex projective manifolds (or more generally orbifolds, if Γ has torsion) whose boundaries are well-behaved. Theorem 1.20. Let Γ be an inﬁnite discrete subgroup of PGL(V ). The following are equivalent: (1) Γ is convex cocompact in P(V ) (Deﬁnition 1.11): it acts convex cocompactly on a nonempty properly convex open subset Ω of P(V ); (2) Γ acts properly discontinuously and cocompactly on a nonempty properly convex set Cbisat ⊂ P(V ) with bisaturated boundary; (3) Γ acts properly discontinuously and cocompactly on a nonempty properly convex set Cstrict ⊂ P(V ) with strictly convex nonideal boundary; (4) Γ acts properly discontinuously and cocompactly on a nonempty prop- 1 erly convex set Csmooth ⊂ P(V ) with strictly convex C nonideal boundary.

In this case Cbisat, Cstrict, Csmooth can be chosen equal, with Ω satisfying orb ΛΩ (Γ) = ∂iCsmooth. Remark 1.21. Given a properly discontinuous and cocompact action of a group Γ on a properly convex set C, Theorem 1.20 interprets the convex cocompactness of Γ in terms of the regularity of C at the nonideal boundary ∂nC, whereas Theorems 1.15 and 1.20 together interpret the strong convex cocompactness of Γ in terms of the regularity of C at both ∂iC and ∂nC. The equivalence (1) ⇔ (3) of Theorem 1.20 states that convex cocompact torsion-free subgroups in P(V ) are precisely the holonomy groups of compact properly convex projective manifolds with strictly convex boundary. Cooper–Long–Tillmann [CLT2] studied the deformation theory of such manifolds (allowing in addition certain types of cusps). They established 12 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

n−1 n Figure 1. The group Z acts diagonally on P(R ) ⊂ n+1 P(R ), where n = 2 (top row) or 3 (bottom row). These actions are convex cocompact; each row shows two convex invariant domains with bisaturated boundary, the domain on the left having also strictly convex C1 nonideal boundary. We show ∂iΩ in red and ∂nΩ in blue. a holonomy principle, which reduces to the following statement in the absence of cusps: the holonomy groups of compact properly convex projective manifolds with strictly convex boundary form an open subset of the representation space of Γ. This result, together with Theorem 1.20, gives the stability property (D) of Theorem 1.17. For the case of divisible convex sets, see [K]. The requirement that Cbisat have bisaturated boundary in condition (2) can be seen as a “coarse” version of the strict convexity of the nonideal boundary in (4): the prototype situation is that of a convex cocompact real hyperbolic manifold, with Cbisat a closed polyhedral neighborhood of the convex core, see the right-hand side of Figure 1. Properly convex sets with bisaturated boundary behave well under a natural duality operation generalizing that of open properly convex sets (see Section 5). The equivalence (1) ⇔ (2) will be used to prove the duality property (A) of Theorem 1.17. When the ideal boundary is closed, both conditions “strictly convex and C1 nonideal boundary” and “bisaturated boundary” are invariant under duality, but it can be easier to build examples of the latter: condition (2) of Theorem 1.20 is convenient in some key arguments of the paper. We note that without some form of convexity requirement on the boundary, properly convex projective manifolds have a poorly-behaved deformation theory: see Remark 3.10.(b). Remark 1.22. If Γ is strongly irreducible (i.e. all ﬁnite-index subgroups of Γ are irreducible) and if the equivalent conditions of Theorem 1.20 hold, then CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 13

orb ΛΩ (Γ) = ∂iCbisat = ∂iCstrict = ∂iCsmooth for any Ω, Cbisat, Cstrict, Csmooth as in conditions (1), (2), (3), and (4) respectively: see Section 4.3. In particular, this set then depends only on Γ. This is not necessarily the case if Γ is not strongly irreducible: see Examples 3.7 and 3.8.

1.8. Convex cocompactness for subgroups of PO(p, q). As mentioned in Remark 1.5.(b), when Γ ⊂ PGL(V ) is irreducible and contained in the subgroup PO(p, q) ⊂ PGL(V ) of projective linear transformations that preserve a nondegenerate symmetric bilinear form h·, ·ip,q of some signature (p, q) on n V = R , a more precise version of Theorem 1.4 was established in [DGK3, Th. 1.11 & Prop. 1.17]. Here we remove the irreducibility assumption on Γ. ∗ p,q p+q For p, q ∈ N , let R be R endowed with a symmetric bilinear form h·, ·ip,q of signature (p, q). The spaces p,q−1 p,q H = {[x] ∈ P(R ) | hx, xip,q < 0} p−1,q p,q and S = {[x] ∈ P(R ) | hx, xip,q > 0} are the projective models for pseudo-Riemannian hyperbolic and spherical geometry respectively. The geodesics of these two spaces are the intersections p,q of the spaces with the projective lines of P(R ). The group PO(p, q) acts p,q−1 p−1,q transitively on H and on S . Multiplying the form h·, ·ip,q by −1 produces a form of signature (q, p) and turns PO(p, q) into PO(q, p) and p−1,q q,p−1 S into a copy of H , so we consider only the pseudo-Riemannian p,q−1 p,q−1 p,q−1 hyperbolic spaces H . We denote by ∂H the boundary of H , namely p,q−1 p,q ∂H = {[x] ∈ P(R ) | hx, xip,q = 0} . p,q−1 We call a subset of H convex (resp. properly convex) if it is convex (resp. p,q p,q properly convex) as a subset of P(R ). Since the straight lines of P(R ) p,q−1 are the geodesics of the pseudo-Riemannian metric on H , convexity in p,q−1 H is an intrinsic notion.

p,q−1 Remarks 1.23. Let C be a closed convex subset of H . (a) If C has nonempty interior, then C is properly convex. Indeed, if C is not p+q−1 properly convex, then it contains a line ` of an aﬃne chart R ⊃ C, and C is a union of lines parallel to ` in that chart. All these lines must p,q−1 p,q−1 be tangent to ∂H at their common endpoint z ∈ ∂H , which implies that C is contained in the hyperplane z⊥ and has empty interior. p,q−1 (b) The ideal boundary ∂iC is the set of accumulation points of C in ∂H . This set contains no projective line segment if and only if it is transverse, ⊥ i.e. y∈ / z for all y 6= z in ∂iC. p,q−1 (c) The nonideal boundary ∂nC = Fr(C) ∩ H is the boundary of C p,q−1 in H in the usual sense. (d) It is easy to see (Lemma 11.6) that C has bisaturated boundary if and p,q−1 only if a segment contained in ∂nC ⊂ H never extends to a full p,q−1 geodesic of H — a form of coarse strict convexity for ∂nC. 14 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

The following definition was studied in [DGK3] for irreducible discrete subgroups Γ. Definition 1.24 ([DGK3, Def. 1.2]). A discrete subgroup Γ of PO(p, q) is p,q−1 H -convex cocompact if it acts properly discontinuously with compact p,q−1 quotient on some closed properly convex subset C of H with nonempty p,q−1 interior whose ideal boundary ∂iC ⊂ ∂H does not contain any nontrivial projective line segment. If Γ is irreducible, then any nonempty Γ-invariant properly convex subset p,q−1 of H has nonempty interior, and so the nonempty interior requirement in Definition 1.24 may simply be replaced by the requirement that C be nonempty. See Section 11.9 for examples. For a word hyperbolic group Γ, we shall say that a representation ρ : p,q Γ → PO(p, q) is P1 -Anosov if it is P1-Anosov as a representation into p+q PGL(R ). We refer to [DGK3] for further discussion of this notion. p+q If the natural inclusion Γ ,→ PO(p, q) is P1 -Anosov, then the boundary p,q−1 ∗ map ξ takes values in ∂H and the hyperplane-valued boundary map ξ satisfies ξ∗(·) = ξ(·)⊥ where z⊥ denotes the orthogonal of z with respect to p,q−1 h·, ·ip,q; the image of ξ is the proximal limit set ΛΓ of Γ in ∂H (Defi- nition 2.2 and Remark 11.1). Following [DGK3, Def. 1.9], we shall say that p,q ΛΓ is negative (resp. positive) if it lifts to a cone of R r {0} on which all inner products h·, ·ip,q of noncollinear points are negative; equivalently (see [DGK3, Lem. 3.2]), any three distinct points of ΛΓ span a linear subspace p,q of R of signature (2, 1) (resp. (1, 2)). With this notation, we prove the following, where Γ is not assumed to be irreducible; for the irreducible case, see [DGK3, Th. 1.11 & Prop. 1.17].

∗ Theorem 1.25. For p, q ∈ N and for an arbitrary inﬁnite discrete subgroup Γ of PO(p, q), the following are equivalent: p+q (1) Γ is strongly convex cocompact in P(R ) (Deﬁnition 1.1); p,q−1 q,p−1 (2) Γ is H -convex cocompact or H -convex cocompact (after identifying PO(p, q) with PO(q, p) as above). The following are also equivalent: p+q p,q−1 (3) Γ is strongly convex cocompact in P(R ) and ΛΓ ⊂ ∂H is negative; p,q−1 (4) Γ is H -convex cocompact; (5) Γ acts convex cocompactly on some nonempty properly convex open p,q−1 subset Ω of H and ΛΓ is transverse; p,q (6) Γ is word hyperbolic, the natural inclusion Γ ,→ PO(p, q) is P1 - p,q−1 Anosov, and ΛΓ ⊂ ∂H is negative. Similarly, the following are equivalent: p+q p,q−1 (7) Γ is strongly convex cocompact in P(R ) and ΛΓ ⊂ ∂H is positive; q,p−1 (8) Γ is H -convex cocompact (after identifying PO(p, q) with PO(q, p)); CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 15

(9) Γ acts convex cocompactly on some nonempty properly convex open q,p−1 subset Ω of H (after identifying PO(p, q) with PO(q, p)) and ΛΓ is transverse; p,q (10) Γ is word hyperbolic, the natural inclusion Γ ,→ PO(p, q) is P1 - p,q−1 Anosov, and ΛΓ ⊂ ∂H is positive. p+q p+q If Γ < PO(p, q) < PGL(R ) is convex cocompact in P(R ) (Defini- tion 1.11) and irreducible, then it is in fact strongly convex cocompact in p+q P(R ). It is not difficult to see that if T is a connected open subset of Hom(Γ, PO(p, q)) p,q p,q−1 consisting entirely of P1 -Anosov representations, and if Λρ(Γ) ⊂ ∂H is negative for some ρ ∈ T , then it is negative for all ρ ∈ T [DGK3, Prop. 3.5]. Therefore Theorem 1.25 implies the following. ∗ Corollary 1.26. For p, q ∈ N and for a discrete group Γ, let T be a con- p,q nected open subset of Hom(Γ, PO(p, q)) consisting entirely of P1 -Anosov p,q−1 q,p−1 representations. If ρ(Γ) is H -convex cocompact (resp. H -convex p,q−1 cocompact) for some ρ ∈ T , then ρ(Γ) is H -convex cocompact (resp. q,p−1 H -convex cocompact) for all ρ ∈ T . p,q−1 It is also not difficult to see that if a closed subset Λ of ∂H is transverse (i.e. z⊥ ∩Λ = {z} for all z ∈ Λ) and connected, then it is negative or positive [DGK3, Prop. 1.10]. Therefore Theorem 1.25 implies the following. Corollary 1.27. Let Γ be a word hyperbolic group with connected bound- ∗ p,q ary ∂∞Γ, and let p, q ∈ N . For any P1 -Anosov representation ρ :Γ → p,q−1 q,p−1 PO(p, q), the group ρ(Γ) is H -convex cocompact or H -convex cocompact (after identifying PO(p, q) with PO(q, p)). p,q−1 p+1 Remark 1.28. In the special case when q = 2 (i.e. H = AdS is the Lorentzian anti-de Sitter space) and Γ is the fundamental group of a closed hyperbolic p-manifold, the equivalence (4) ⇔ (6) of Theorem 1.25 follows from work of Mess [Me] for p = 2 and Barbot–Mérigot [BM] for p ≥ 3. p,q−1 As a consequence of Theorem 1.20, we obtain characterizations of H - convex cocompactness where the assumption on the ideal boundary ∂iC ⊂ p,q−1 ∂H in Definition 1.24 is replaced by various regularity conditions on the p,q−1 nonideal boundary ∂nC ⊂ H . ∗ Theorem 1.29. For any p, q ∈ N and any arbitrary infinite discrete subgroup Γ of PO(p, q), the following are equivalent: p,q−1 (1) Γ is H -convex cocompact: it acts properly discontinuously and p,q−1 cocompactly on a closed convex subset C of H with nonempty interior whose ideal boundary ∂iC does not contain any nontrivial projective line segment; (2) Γ acts properly discontinuously and cocompactly on a nonempty closed p,q−1 p,q−1 convex subset Cbisat of H whose boundary ∂nCbisat in H does p,q−1 not contain any infinite geodesic line of H ; 16 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

(3) Γ acts properly discontinuously and cocompactly on a nonempty closed p,q−1 p,q−1 convex subset Cstrict of H whose boundary ∂nCstrict in H is strictly convex; (4) Γ acts properly discontinuously and cocompactly on a nonempty closed p,q−1 p,q−1 convex set Csmooth of H whose boundary ∂nCsmooth in H is strictly convex and C1.

In this case, C, Cbisat, Cstrict, and Csmooth may be taken equal. 1.9. Organization of the paper. Section 2 contains reminders about properly convex domains in projective space, the Cartan decomposition, and Anosov representations. In Section 3 we establish some facts about actions on convex subsets of P(V ), and give a few basic (non-)examples for convex cocompactness. In Section 4 we prove the equivalence (1) ⇔ (2) of Theorem 1.20. In Section 5 we study a notion of duality and establish Theorem 1.17.(A). Sections 6 to 9 are devoted to the proofs of the main The- orems 1.15 and 1.20, which contain Theorem 1.4. In Section 10 we establish properties (B)–(F) of Theorem 1.17. In Section 11 we prove Theorems 1.25 p,q−1 and 1.29 on H -convex cocompactness. Section 12 is devoted to examples, in particular Proposition 1.7. Finally, in Appendix A we collect a few open questions on convex cocompact groups in P(V ). Acknowledgements. We are grateful to Yves Benoist, Sam Ballas, Gye- Seon Lee, Arielle Leitner, and Ludovic Marquis for helpful and inspiring discussions, as well as to Thierry Barbot for his encouragement. We thank Daryl Cooper for a useful reference. The ﬁrst-named author thanks Steve Kerckhoﬀ for discussions related to Example 10.11 from several years ago, which proved valuable for understanding the general theory.

2. Reminders 2.1. Properly convex domains in projective space. Let Ω be a properly convex open subset of P(V ), with boundary ∂Ω. Recall the Hilbert metric dΩ on Ω: 1 d (y, z) := log [a, y, z, b] Ω 2 1 for all distinct y, z ∈ Ω, where [·, ·, ·, ·] is the cross-ratio on P (R), normalized so that [0, 1, z, ∞] = z, and where a, b are the intersection points of ∂Ω with the projective line through y and z, with a, y, z, b in this order. The metric space (Ω, dΩ) is proper (closed balls are compact) and complete, and the group Aut(Ω) := {g ∈ PGL(V ) | g · Ω = Ω} acts on Ω by isometries for dΩ. As a consequence, any discrete subgroup of Aut(Ω) acts properly discontinuously on Ω. Let V ∗ be the dual vector space of V . By definition, the dual convex set of Ω is ∗ ∗ Ω := P ` ∈ V | `(x) < 0 ∀x ∈ Ωe , CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 17 where Ωe is the closure in V r {0} of an open convex cone of V lifting Ω. The ∗ ∗ set Ω is a properly convex open subset of P(V ) which is preserved by the ∗ dual action of Aut(Ω) on P(V ). Straight lines (contained in projective lines) are always geodesics for the Hilbert metric dΩ. When Ω is not strictly convex, there may be other geodesics as well. However, a biinfinite geodesic of (Ω, dΩ) always has well- defined, distinct endpoints in ∂Ω, see [DGK3, Lem. 2.6].

2.2. Cartan decomposition. The group G˜ = GL(V ) admits the Cartan decomposition G˜ = K˜ exp(a˜+)K˜ where K˜ = O(n) and + a˜ := {diag(t1, . . . , tn) | t1 ≥ · · · ≥ tn}.

This means that any g ∈ G˜ may be written g = k1 exp(a)k2 for some k1, k2 ∈ + K˜ and a unique a = diag(t1, . . . , tn) ∈ a˜ ; for any 1 ≤ i ≤ n, the real number ti is the logarithm µi(g) of the i-th largest singular value of g. This induces a Cartan decomposition G = K exp(a+)K of G = PGL(V ), with K = PO(n) + + and a = a˜ /R, and for any 1 ≤ i < j ≤ n a map + (2.1) µi − µj : G −→ R .

If k · kV denotes the operator norm associated with the standard Euclidean n norm on V = R invariant under K˜ = O(n), then for all g ∈ G we have −1 (2.2) (µ1 − µn)(g) = log kgkV kg kV .

2.3. Proximality in projective space. We shall use the following classical terminology.

∗ Deﬁnition 2.1. An element g ∈ PGL(V ) is proximal in P(V ) (resp. P(V )) ∗ if it admits a unique attracting ﬁxed point in P(V ) (resp. P(V )). Equiva- lently, g has a unique complex eigenvalue of maximal (resp. minimal) mod- ulus. This eigenvalue is necessarily real.

For any g ∈ GL(V ), we denote by λ1(g) ≥ λ2(g) ≥ · · · ≥ λn(g) the logarithms of the moduli of the complex eigenvalues of g. For any 1 ≤ i < j ≤ n, this induces a function + (2.3) λi − λj : PGL(V ) −→ R . ∗ Thus, an element g ∈ PGL(V ) is proximal in P(V ) (resp. P(V )) if and only if (λ1 − λ2)(g) > 0 (resp. (λn−1 − λn)(g) > 0). We shall use the following terminology.

Definition 2.2. Let Γ be a discrete subgroup of PGL(V ). The proximal limit set of Γ in P(V ) is the closure ΛΓ of the set of attracting fixed points of elements of Γ which are proximal in P(V ). Remark 2.3. When Γ is an irreducible discrete subgroup of PGL(V ) containing at least one proximal element, the proximal limit set ΛΓ was first 18 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL studied in [Gu, B1, B2]. In that setting, the action of Γ on ΛΓ is minimal (i.e. any orbit is dense), and ΛΓ is contained in any nonempty, closed, Γ-invariant subset of P(V ), by [B2, Lem. 2.5].

2.4. Anosov representations. Let P1 (resp. Pn−1) be the stabilizer in G = PGL(V ) of a line (resp. hyperplane) of V ; it is a maximal proper parabolic subgroup of G, and G/P1 (resp. G/Pn−1) identifies with P(V ) (resp. with ∗ the dual projective space P(V )). As in the introduction, we shall think ∗ of P(V ) as the space of projective hyperplanes in P(V ). The following is not the original definition from [L, GW3], but an equivalent characterization taken from [GGKW, Th. 1.7 & Rem. 4.3.(c)]. Definition 2.4. Let Γ be a word hyperbolic group. A representation ρ : Γ → G = PGL(V ) is P1-Anosov if there exist two continuous, ρ-equivariant ∗ ∗ boundary maps ξ : ∂∞Γ → P(V ) and ξ : ∂∞Γ → P(V ) such that ∗ ∗ (A1) ξ and ξ are compatible, i.e. ξ(η) ∈ ξ (η) for all η ∈ ∂∞Γ; ∗ ∗ 0 0 (A2) ξ and ξ are transverse, i.e. ξ(η) ∈/ ξ (η ) for all η 6= η in ∂∞Γ; (A3)’ ξ and ξ∗ are dynamics-preserving and there exist c, C > 0 such that for any γ ∈ Γ,

(λ1 − λ2)(ρ(γ)) ≥ c `Γ(γ) − C,

where `Γ :Γ → N is the translation length function of Γ in its Cayley graph (for some ﬁxed choice of ﬁnite generating subset).

In condition (A3)’ we use the notation λ1 − λ2 from (2.3). By dynamics- preserving we mean that for any γ ∈ Γ of infinite order, the element ρ(γ) ∈ G ∗ is proximal and ξ (resp. ξ ) sends the attracting fixed point of γ in ∂∞Γ to ∗ the attracting fixed point of ρ(γ) in P(V ) (resp. P(V )). In particular, the ∗ set ξ(∂∞Γ) (resp. ξ (∂∞Γ)) is the proximal limit set (Definition 2.2) of ρ(Γ) ∗ in P(V ) (resp. P(V )). By [GW3, Prop. 4.10], if ρ(Γ) is irreducible then condition (A3)’ is automatically satisfied as soon as (A1) and (A2) are, but this is not true in general (see [GGKW, Ex. 7.15]). If Γ is not elementary (i.e. not cyclic up to finite index), then the action of Γ on ∂∞Γ is minimal, i.e. every orbit is dense; therefore the action of Γ ∗ on ξ(∂∞Γ) and ξ (∂∞Γ) is also minimal. We shall use a related characterization of Anosov representations, which we take from [GGKW, Th. 1.3]; it also follows from [KLPa]. Fact 2.5. Let Γ be a word hyperbolic group. A representation ρ :Γ → G = PGL(V ) is P1-Anosov if there exist two continuous, ρ-equivariant boundary ∗ ∗ maps ξ : ∂∞Γ → P(V ) and ξ : ∂∞Γ → P(V ) satisfying conditions (A1) – (A2) of Definition 2.4, as well as (A3)” ξ and ξ∗ are dynamics-preserving and

(µ1 − µ2)(ρ(γ)) −→ +∞, |γ|→+∞

where | · |Γ :Γ → N is the word length function of Γ (for some ﬁxed choice of ﬁnite generating subset). CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 19

In condition (A3)” we use the notation µ1 − µ2 from (2.1). By construction, the image of the boundary map ξ : ∂∞Γ → P(V ) (resp. ∗ ∗ ξ : ∂∞Γ → P(V )) of a P1-Anosov representation ρ :Γ → PGL(V ) is the closure of the set of attracting ﬁxed points of proximal elements of ρ(Γ) in ∗ P(V ) (resp. P(V ). Here is a useful alternative description. Fact 2.6 ([GGKW, Th. 1.3 & 5.3]). Let Γ be a word hyperbolic group and ρ :Γ → PGL(V ) a P1-Anosov representation with boundary maps ξ : ∂∞Γ → ∗ ∗ P(V ) and ξ : ∂∞Γ → P(V ). Let (γm)m∈N be a sequence of elements of Γ converging to some η ∈ ∂∞Γ. For any m, choose km ∈ K such that ρ(γm) ∈ + ∗ km exp(a )K (see Section 2.2). Then, writing [en] := P(span(e1, . . . , en−1)), ξ(η) = limm→+∞ km · [e1], ∗ ∗ ξ (η) = limm→+∞ km · [en]. In particular, the image of ξ is the set of accumulation points in P(V ) of + ∗ the set {kρ(γ) · [e1] | γ ∈ Γ} where γ ∈ kρ(γ) exp(a )K; the image of ξ is the ∗ ∗ set of accumulation points in P(V ) of {kρ(γ) · [en] | γ ∈ Γ}. Here is an easy consequence of Fact 2.5. Remark 2.7. Let Γ be a word hyperbolic group, Γ0 a subgroup of Γ, and ρ :Γ → PGL(V ) a representation. 0 • If Γ has ﬁnite index in Γ, then ρ is P1-Anosov if and only if its 0 restriction to Γ is P1-Anosov. 0 • If Γ is quasi-isometrically embedded in Γ and if ρ is P1-Anosov, then 0 the restriction of ρ to Γ is P1-Anosov.

3. Basic facts: actions on convex subsets of P(V ) In this section we collect a few useful facts, and give some basic examples and non-examples of convex cocompact actions.

3.1. Divergence for actions on properly convex cones. We shall often use the following observation. Remark 3.1. Let Γ be a discrete subgroup of PGL(V ) preserving a nonempty properly convex open subset Ω of P(V ). There is a unique lift Γˆ of Γ to SL±(V ) that preserves a properly convex cone Ωe of V lifting Ω. Here we say that a convex open cone of V is properly convex if its projection to P(V ) is properly convex in the sense of Section 1.1. The following observation will be useful in Sections 7 and 10. Lemma 3.2. Let Γˆ be a discrete subgroup of SL±(V ) preserving a properly convex open cone Ωe in V . For any sequence (γm)m∈N of pairwise distinct ˆ elements of Γ and any nonzero vector x ∈ Ωe, the sequence (γm · x)m∈N goes to inﬁnity as m → +∞. This divergence is uniform as x varies in a compact set K ⊂ Ωe. 20 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Proof. Fix a compact subset K of Ωe. Let ` ∈ V ∗ be a linear form which takes positive values on the closure of Ωe. The set Ωe ∩ {` = 1} is bounded, with compact boundary B in V . By compactness of K and B, we can ﬁnd 0 < ε < 1 such that for any x ∈ K and y ∈ B the line through x/`(x) and y intersects B in a point y0 6= y such that x/`(x) = ty + (1 − t)y0 for some t ≥ ε. For any m ∈ N, we then have `(γm · x)/`(x) ≥ ε `(γm · y), and since this holds for any y ∈ B we obtain

`(γm · x) ≥ κ max(` ◦ γm) B where κ := ε minK(`) > 0. Thus it is sufficient to see that the maximum of ` ◦ γm over B tends to infinity with m. By convexity, it is in fact sufficient to see that the maximum of ` ◦ γm over Ωe ∩ {` < 1} tends to infinity with m. This follows from the fact that the set Ωe ∩ {` < 1} is open, that the operator norm of γm ∈ Γˆ ⊂ End(V ) goes to +∞, and that ` is equivalent to any norm of V in restriction to Ωe. The following immediate corollary will be used in Section 11. ∗ Corollary 3.3. For p, q ∈ N , let Γ be an infinite discrete subgroup of p,q−1 PO(p, q) preserving a properly convex open subset Ω of H . Then the orb p,q−1 full orbital limit set ΛΩ (Γ) is contained in ∂H . 3.2. Comparison between the Hilbert and Euclidean metrics. The following will be used later in this section and in the proofs of Lemmas 4.1 and 10.6. Lemma 3.4. Let Ω ⊂ P(V ) be open, properly convex, contained in a Eu- n−1 clidean affine chart (R , dEuc) of P(V ). If R > 0 is the diameter of Ω in n−1 (R , dEuc), then the natural inclusion defines an R/2-Lipschitz map n−1 (Ω, dΩ) −→ (R , dEuc). Proof. We may assume n = 2 up to restricting to one line of Ω, and R = 2 1 up to scaling. The arclength parametrization of Ω ' H ' R is then given by the 1-Lipschitz map R → (−1, 1) sending t to tanh(t).

Corollary 3.5. Let Ω be a properly convex open subset of P(V ). Let (ym)m∈N and (zm)m∈N be two sequences of points of Ω, and y ∈ ∂Ω. If ym → y and if dΩ(ym, zm) → 0, then zm → y. 3.3. Closed ideal boundary. The following observation will be used in Sections 4, 5, and 9. Lemma 3.6. Let Γ be a discrete subgroup of PGL(V ) and C a Γ-invariant convex subset of P(V ). Suppose the action of Γ on C is properly discontinuous and cocompact. Then ∂iC is closed in P(V ).

Proof. Let (zm)m∈N be a sequence of points of ∂iC converging to some z∞ ∈ Fr(C). Suppose for contradiction that z∞ ∈/ ∂iC, so that z∞ ∈ ∂nC. For each m, let (zm,k)k∈N be a sequence of points of C converging to zm as k → +∞. CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 21

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3 Figure 2. The subgroup Γ ⊂ PGL(R ) of all diagonal matrices with each entry a power of t = 2 divides a triangle Ω 3 in P(R ) with vertices the standard basis. Shown are two Γ-orbits with diﬀerent accumulation sets in ∂Ω.

Let D ⊂ C be a compact fundamental domain for the action of Γ. For any m, k ∈ N, there exists γm,k ∈ Γ such that zm,k ∈ γm,k ·D. Let (zm,km )m∈N be a sequence converging to z∞. Then the set {z∞} ∪ {zm,km }m∈N is compact and intersects all γm,km ·D, contradicting properness. 3.4. Examples and non-examples. The following basic examples are de- signed to make the notion of convex cocompactness in P(V ) more concrete, and to point out some subtleties.

n n−1 Example 3.7. For V = R with n ≥ 2, let Γ ' Z be the discrete subgroup of PGL(V ) of diagonal matrices whose entries are powers of some ﬁxed t > 1; it is not word hyperbolic. The hyperplanes

Hk = P(span(e1, . . . , ek−1, ek+1, . . . , en)) n−1 for 1 ≤ k ≤ n cut P(V ) into 2 properly convex open connected components Ω, which are not strictly convex (see Figure 2). The group Γ acts properly discontinuously and cocompactly on each of them, hence is convex cocompact in P(V ) (see Example 1.13). We note that in Example 3.7 the set of accumulation points of one Γ-orbit of Ω depends on the choice of orbit (see Figure 2), but the full orbital limit orb set ΛΩ (Γ) = ∂Ω does not; the proximal limit set ΛΓ (Deﬁnition 2.2) is the orb set of extremal points of the full orbital limit set ΛΩ (Γ). Here is a noncommutative version of Example 3.7, containing it as a subgroup of ﬁnite index.

n Example 3.8. Let V = R with n ≥ 2. For any 1 ≤ i ≤ n − 1, let si ∈ GL(V ) switch ei and ei+1 and keep all other ej fixed. Let sn ∈ GL(V ) send −1 (e1, en) to (te1, t en) for some t > 1, and keep all other ej fixed. The discrete subgroup of PGL(V ) generated by s1, . . . , sn (an affine Coxeter group of type An) is irreducible but not strongly irreducible. It still preserves and acts cocompactly on each of the 2n−1 properly convex connected components of Example 3.7, hence it is convex cocompact in P(V ). 22 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

3 Examples 3.9. Let V = R , with standard basis (e1, e2, e3). Let Γ be a cyclic group generated by an element γ ∈ PGL(V ). a 0 0 (1) Suppose γ = 0 b 0 where a > b > c > 0. Then γ has attracting fixed 0 0 c point at [e1] and repelling fixed point at [e3]. There is a Γ-invariant open properly convex neighborhood Ω of an open segment ([e1], [e3]) orb connecting [e1] to [e3]. The full orbital limit set ΛΩ (Γ) is just {[e1], [e3]} cor and its convex hull CΩ (Γ) = ([e1], [e3]) has compact quotient by Γ (a circle). Thus Γ is convex cocompact in P(V ). a 0 0 (2) Suppose γ = 0 b 0 where a > b > 0. Any Γ-invariant properly convex 0 0 b open set Ω is a triangle with tip [e1] and base a segment I of the line P(span(e2, e3)). For any z ∈ Ω, the Γ-orbit of z has two accumulation points, namely [e1] and the intersection of I with the line through [e1], z. orb Therefore, the full orbital limit set ΛΩ (Γ) consists of the point [e1] and cor orb the interior of I, hence the convex hull CΩ (Γ) of ΛΩ (Γ) is the whole of Ω, and Γ does not act cocompactly on it. Thus Γ is not convex cocompact in P(V ). However, Γ is naively convex cocompact in P(V ) (Definition 1.9): it acts cocompactly on the convex hull C in Ω of {[e1]} and of any closed segment I0 contained in the interior of I. The quotient Γ\C is a closed convex projective annulus (or a circle if I0 is reduced to a singleton). 2 0 0 (3) Suppose γ = 0 1 t where t > 0. Then there exist nonempty Γ- 0 0 1 invariant properly convex open subsets of P(V ), for instance x /t Ωs = P({(x1, x2, 1) | x1 > s 2 2 }) for any s > 0. However, for any orb orb such set Ω, we have ΛΩ (Γ) = {[e1], [e2]}, and the convex hull of ΛΩ (Γ) cor in Ω is contained in ∂Ω (see [Ma, Prop. 2.13]). Hence CΩ (Γ) is empty and Γ is not convex cocompact in P(V ). It is an easy exercise to check that Γ is not even naively convex cocompact in P(V ) (Definition 1.9). a 0 0 (4) Suppose γ = 0 b cos θ −b sin θ where a > b > 0 and 0 < θ ≤ π. Then 0 b sin θ b cos θ Γ does not preserve any nonempty properly convex open subset of P(V ) (see [Ma, Prop. 2.4]). From these examples we observe the following.

Remarks 3.10. (a) Convex cocompactness in P(V ) is not a closed condition in general. Indeed, Example 3.9.(2), which is not convex cocompact, is a limit of Example 3.9.(1), which is convex cocompact. (b) Naive convex cocompactness (Deﬁnition 1.9) is not an open condition (even if we require the cocompact convex subset C ⊂ Ω to have nonempty interior). Indeed, Example 3.9.(2), which satisﬁes the condition, is a limit of both 3.9.(3) and 3.9.(4), which do not. Here is a slightly more complicated example of a discrete subgroup of PGL(V ) which is naively convex cocompact but not convex cocompact in P(V ). CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 23

Example 3.11. Let Γ1 be a convex cocompact subgroup of SO(2, 1) ⊂ 3 1 ∗ 1 GL(R ), let Γ2 ' Z be a discrete subgroup of GL(R ) ' R acting on R 3 1 by scaling, and let Γ = Γ1 × Γ2 ⊂ GL(R ⊕ R ). Any Γ-invariant properly 3 1 convex open subset Ω of P(R ⊕ R ) is a cone Ω1 ? z with base some Γ1- 3 3 1 invariant properly convex open subset Ω1 of P(R ) ⊂ P(R ⊕ R ) and tip 1 3 1 orb z := P(R ) ⊂ P(R ⊕R ). The full orbital limit set ΛΩ (Γ) contains both {z} cor and the full base Ω1, hence CΩ (Γ) is equal to Ω. Therefore Γ acts convex cocompactly on Ω if and only if Γ1 acts cocompactly on Ω1. If Γ1 is not cocompact in SO(2, 1), then Γ1 does not act cocompactly on any nonempty 3 properly convex open subset of P(R ), hence Γ is not convex cocompact in 3 1 2 P(R ⊕ R ). Nevertheless, in the case that Ω1 = H is the round disc, Γ acts cocompactly on the closed convex subcone C ⊂ Ω with base the convex hull 2 in H of the limit set of Γ1. In further work [DGK5], we shall describe an example of an irreducible 4 discrete subgroup of PGL(R ) which is naively convex cocompact but not convex cocompact in P(V ), namely a free group containing Example 3.9.(2) as a free factor.

4. Convex sets with bisaturated boundary In this section we prove the equivalence (1) ⇔ (2) of Theorem 1.20. In Sections 7 and 8, we will use these conditions interchangeably for Defini- tion 1.11. We prove (Corollaries 4.2 and 4.4) that when these conditions hold, sets Ω as in condition (1) and Cbisat as in condition (2) may be chosen orb so that the full orbital limit set ΛΩ (Γ) of Γ in Ω coincides with the ideal boundary ∂iCbisat of Cbisat. In Section 4.3, we show that if Γ is strongly ir- orb reducible and satisfies conditions (1) and (2), then ΛΩ (Γ) and ∂iCbisat are uniquely determined (Fact 4.7 and Proposition 4.8). cor Recall that CΩ (Γ) denotes the convex core of Ω for Γ, i.e. the convex hull orb of ΛΩ (Γ) in Ω. 4.1. Ideal boundaries of invariant convex sets. Let us prove the implication (1) ⇒ (2) of Theorem 1.20. It relies on the following lemma, which will be used many times in the paper. Lemma 4.1. Let Γ be a discrete subgroup of PGL(V ) and Ω a nonempty Γ-invariant properly convex open subset of P(V ). (1) If Γ acts cocompactly on some closed convex subset C of Ω, then for orb any z ∈ ∂iC the open stratum of ∂Ω at z is contained in ΛΩ (Γ); in orb cor particular, ∂iC ⊂ ΛΩ (Γ). If moreover C contains CΩ (Γ), then cor orb ∂iC = ∂iCΩ (Γ) = ΛΩ (Γ) orb cor and the set ΛΩ (Γ) is closed in P(V ), the set CΩ (Γ) is closed in Ω, and the action of Γ on Ω is convex cocompact (Definition 1.11). cor (2) If the action of Γ on Ω is convex cocompact, then CΩ (Γ) is a minimal nonempty Γ-invariant closed properly convex subset of Ω. 24 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

(3) Let C0 ⊂ C1 be two nonempty closed properly convex subsets of Ω on which Γ acts cocompactly. If ∂iC1 = ∂iC0 and if C1 contains a neighborhood of C0, then C1 has bisaturated boundary.

In (1), we call open stratum of ∂Ω at y the union Fy of {y} and of all open segments of ∂Ω containing y. It is a properly convex subset of ∂Ω which is open in the projective subspace P(Wy) that it spans. In particular, we can consider the Hilbert metric dFy on Fy seen as a properly convex open subset of P(Wy). The set ∂Ω is the disjoint union of its open strata.

Proof. (1) Suppose that Γ acts cocompactly on a closed convex subset C 0 of Ω. Let z ∈ ∂Ω belong to the stratum of some z ∈ ∂iC. Let us show 0 orb that z ∈ ΛΩ (Γ). Let (a, b) be a maximal open interval of ∂Ω containing z and z0. Choose y ∈ C. The rays (y, z) and (y, z0) remain at bounded distance 0 0 in (Ω, dΩ). Therefore we can find points zm ∈ (y, z) and zm ∈ (y, z ) such 0 0 0 that zm → z and zm → z and dΩ(zm, zm) remains bounded. Since Γ acts N −1 cocompactly on C, there exists (γm) ∈ Γ such that γm · zm remains in −1 −1 0 0 some compact subset of C ⊂ Ω. Since dΩ(γm · zm, γm · zm) = dΩ(zm, zm) −1 0 remains bounded, γm · zm also remains in some compact subset of Ω. Up to −1 0 passing to a subsequence, we may assume that γm · zm converges to some 0 0 0 0 −1 0 0 0 y ∈ Ω. Then dΩ(γm · y , zm) = dΩ(y , γm · zm) → 0, and so γm · y → z by 0 orb orb Corollary 3.5. Thus z ∈ ΛΩ (Γ). In particular, ∂iC ⊂ ΛΩ (Γ). If C contains cor cor orb cor orb CΩ (Γ), then ∂iC ⊃ ∂iCΩ (Γ) ⊃ ΛΩ (Γ), and so ∂iC = ∂iCΩ (Γ) = ΛΩ (Γ). orb In particular, ΛΩ (Γ) is closed in P(V ) by Lemma 3.6, and so its convex hull cor CΩ (Γ) is closed in Ω, and compact modulo Γ because C is. (2) Suppose the action of Γ on Ω is convex cocompact. Consider a cor nonempty closed convex Γ-invariant subset C of Ω contained in CΩ (Γ); cor cor let us prove that C = CΩ (Γ). By definition of CΩ (Γ), it is sufficient to orb orb prove that ∂iC = ΛΩ (Γ), or equivalently that ∂iC ∩ F = ΛΩ (Γ) ∩ F for all open strata F of ∂Ω. We now fix such a stratum F . Since the action cor of Γ on CΩ (Γ) is cocompact, there exists R > 0 such that any point of cor CΩ (Γ) lies at distance ≤ R from a point of C for the Hilbert metric dΩ. It orb is sufficient to prove that the set ΛΩ (Γ) ∩ F is contained in the uniform R-neighborhood of ∂iC ∩ F for the Hilbert metric dF on F ; indeed, since orb ∂iC ∩ F is a closed convex subset of F and since ΛΩ (Γ) ∩ F = F by (1), we orb orb then obtain ∂iC ∩ F = ΛΩ (Γ) ∩ F . Consider a point y ∈ ΛΩ (Γ) ∩ F . Let us orb cor find a point z ∈ ∂iC ∩F such that dF (y, z) ≤ R. Since y ∈ ΛΩ (Γ) ⊂ CΩ (Γ), cor we can find a sequence (ym) of points of CΩ (Γ) converging to y. For any m, there exists zm ∈ C such that dΩ(ym, zm) ≤ R; let am, bm ∈ ∂Ω be such that am, ym, zm, bm are aligned in this order. Up to passing to a subsequence, we may assume that am, ym, zm, bm converge respectively to a, y, z, b ∈ ∂Ω, with z ∈ ∂iC. If y = z, then the proof is finished. Suppose y 6= z, so that a, y, z, b 2R lie in F . The cross ratios [am, ym, zm, bm] ≤ e converge to the cross ratio [a, y, z, b], hence [a, y, z, b] ≤ e2R and a, y, z, b are pairwise distinct and CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 25

cor dF (y, z) ≤ d(a,b)(y, z) ≤ R. Thus ∂iCΩ (Γ) ∩ F is contained in the uniform R-neighborhood of ∂iC ∩ F , which completes the proof of (2). (3) Suppose that ∂iC1 = ∂iC0 and that C1 contains a neighborhood of C0. By cocompactness, there exists ε > 0 such that any point of ∂nC1 is at dΩ- distance ≥ ε from C0. Let H be a supporting hyperplane of C1 and suppose for contradiction that H contains both a point w ∈ ∂nC1 and a point y ∈ ∂iC1. Since ∂iC1 is closed in P(V ) by Lemma 3.6, we may assume without loss of generality that the interval [w, y) is contained in ∂nC1. Consider a sequence zm ∈ [w, y) converging to y. Since Γ acts cocompactly on C1, there is a sequence (γm) ∈ ΓN such that γm · zm remains in some compact subset of C1. Up to taking a subsequence, we may assume that γm · w → w∞ and γm·y → y∞ and γm·zm → z∞ for some w∞, y∞, z∞ ∈ C1. We have y∞ ∈ ∂iC1 since ∂iC1 is closed and w∞ ∈ ∂iC1 since the action of Γ on C1 is properly discontinuous, and z∞ ∈ ∂nC1 since γm · zm remains in some compact subset of C1. The point z∞ ∈ ∂nC1 belongs to the convex hull of {w∞, y∞} ⊂ ∂iC0, hence z∞ ∈ C0: contradiction since z∞ ∈ ∂nC1 is at distance ≥ ε from C0. The implication (1) ⇒ (2) of Theorem 1.20 is contained in the following consequence of Lemma 4.1. Corollary 4.2. Let Γ be an infinite discrete subgroup of PGL(V ) acting convex cocompactly (Definition 1.11) on some nonempty properly convex open cor subset Ω of P(V ). Let C be a closed uniform neighborhood of CΩ (Γ) in (Ω, dΩ). Then C is a properly convex subset of P(V ) with bisaturated boundary on which Γ acts properly discontinuously and cocompactly. Moreover, ∂iC = cor orb ∂iCΩ (Γ) = ΛΩ (Γ). Proof. The set C is properly convex by [Bu, (18.12)]. The group Γ acts properly discontinuously and cocompactly on C since it acts properly dis- cor continuously on Ω and cocompactly on CΩ (Γ). By Lemma 4.1.(1), we have orb ∂iC = ΛΩ (Γ). By Lemma 4.1.(3), the set C has bisaturated boundary. 4.2. Proper cocompact actions on convex sets with bisaturated boundary. In this section we prove the implication (2) ⇒ (1) of Theo- rem 1.20. We first establish the following general observations. Lemma 4.3. Let C be a properly convex subset of P(V ) with bisaturated boundary. Assume C is not closed in P(V ). Then (1) C has nonempty interior Int(C); (2) the convex hull of ∂iC in C is contained in Int(C) whenever ∂iC is closed in P(V ). Proof. (1) If the interior of C were empty, then C would be contained in a hyperplane. Since C has bisaturated boundary, C would be equal to its ideal boundary (hence empty) or to its nonideal boundary (hence closed). (2) Since C has bisaturated boundary, every supporting hyperplane of C at a point z ∈ ∂nC misses ∂iC, hence by the compactness assumption on ∂iC there is a hyperplane strictly separating z from ∂iC (in an affine chart 26 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL containing C). It follows that the convex hull of ∂iC in C does not meet ∂nC, hence is contained in Int(C). The implication (2) ⇒ (1) of Theorem 1.20 is contained in the following consequence of Lemmas 4.1 and 4.3. Corollary 4.4. Let Γ be an infinite discrete subgroup of PGL(V ) acting properly discontinuously and cocompactly on a nonempty properly convex subset C of P(V ) with bisaturated boundary. Then Ω := Int(C) is nonempty and orb Γ acts convex cocompactly (Definition 1.11) on Ω. Moreover, ΛΩ (Γ) = ∂iC. Proof. Since Γ is infinite, C is not closed in P(V ), and so Ω = Int(C) is nonempty by Lemma 4.3.(1). The set ∂iC is closed in P(V ) by Lemma 3.6, and so the convex hull C0 of ∂iC in C is closed in C and contained in Ω by Lemma 4.3.(2). The action of Γ on C0 is still cocompact. Since Γ acts orb cor properly discontinuously on C, the set ΛΩ (Γ) is contained in ∂iC, and CΩ (Γ) is contained in C. By Lemma 4.1.(1), the group Γ acts convex cocompactly orb on Ω and ΛΩ (Γ) = ∂iC0 = ∂iC. 4.3. Maximal invariant convex sets. The following was first observed by n Benoist [B2, Prop. 3.1] for irreducible discrete subgroups of PGL(R ). Here we do not make any irreducibility assumption. Proposition 4.5. Let Γ be a discrete subgroup of PGL(V ) preserving a nonempty properly convex open subset Ω of P(V ) and containing a proximal ∗ element. Let ΛΓ (resp. ΛΓ) be the proximal limit set of Γ in P(V ) (resp. ∗ P(V )) (Definition 2.2). Then ∗ ∗ (1) ΛΓ (resp. ΛΓ) is contained in the boundary of Ω (resp. its dual Ω ); (2) more specifically, Ω and ΛΓ lift to cones Ωe and ΛeΓ of V r {0} with ∗ ∗ Ωe properly convex containing ΛeΓ in its boundary, and Ω and ΛΓ lift ∗ ∗ ∗ ∗ to cones Ωe and ΛeΓ of V r {0} with Ωe properly convex containing ∗ ∗ ΛeΓ in its boundary, such that `(x) ≥ 0 for all x ∈ ΛeΓ and ` ∈ ΛeΓ; ∗ (3) for ΛeΓ as in (2), the set ∗ Ωmax = P({x ∈ V | `(x) > 0 ∀` ∈ ΛeΓ}) S ∗ is the unique connected component of (V ) ∗ ∗ z containing Ω; P r z ∈ΛΓ it is Γ-invariant, convex, and open in P(V ); any Γ-invariant properly 0 convex open subset Ω of P(V ) containing Ω is contained in Ωmax. + Proof. (1) Let γ ∈ Γ be proximal in P(V ), with attracting fixed point zγ − and complementary γ-invariant hyperplane Hγ . Since Ω is open, there exists − m + + y ∈ Ω r Hγ . We then have γ · y → zγ , and so zγ ∈ ∂Ω since the action of ∗ ∗ Γ on Ω is properly discontinuous. Thus ΛΓ ⊂ ∂Ω. Similarly, ΛΓ ⊂ ∂Ω . (2) The set Ω lifts to a properly convex cone Ωe of V r {0}, unique up to global sign. This determines a cone ΛeΓ of V r {0} lifting ΛΓ ⊂ ∂Ω and ∗ ∗ contained in the boundary of Ωe. By definition, Ω is the projection to P(V ) CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 27 of the dual cone ∗ ∗ Ωe := ` ∈ V r {0} | `(x) > 0 ∀x ∈ Ωe r {0} . ∗ ∗ ∗ ∗ This cone determines a cone ΛeΓ of V r {0} lifting ΛΓ ⊂ ∂Ω and contained ∗ ∗ in the boundary of Ωe . By construction, `(x) ≥ 0 for all x ∈ ΛeΓ and ` ∈ ΛeΓ. ∗ (3) The set Ωmax := P({x ∈ V | `(x) > 0 ∀` ∈ ΛeΓ}) is a connected S ∗ component of (V ) ∗ ∗ z . It is convex and it contains Ω. It is Γ- P r z ∈ΛΓ ∗ invariant because Ω is. It is open in P(V ) by compactness of ΛΓ. Moreover, 0 any Γ-invariant properly convex open subset Ω of P(V ) containing Ω is 0 ∗ ∗ ∗ ∗ 0∗ contained in Ωmax: indeed, Ω cannot meet z for z ∈ ΛΓ since ΛΓ ⊂ ∂Ω by (1). Remark 4.6. In the context of Proposition 4.5, when Γ preserves a nonempty properly convex open subset Ω of P(V ) but Γ is not irreducible, the following may happen: (a) Γ may not contain any proximal element in P(V ): this is the case if Γ ⊂ PGL(V 0 ⊕ V 0) is the image of a diagonal embedding of a discrete 0 ± 0 0 group Γˆ ⊂ SL (V ) preserving a properly convex open set in P(V ); ∗ (b) assuming that Γ contains a proximal element in P(V ) (so ΛΓ, ΛΓ 6= ∅), the set Ωmax of Proposition 4.5.(3) may fail to be properly convex: this is the case if Γ is a convex cocompact subgroup of SO(2, 1)0, embedded 4 3 into PGL(R ) where it preserves the properly convex open set H ⊂ 4 P(R ); (c) even if Ωmax is properly convex, it may not be the unique maximal Γ-invariant properly convex open set in P(V ): indeed, there may be S ∗ multiple components of (V ) ∗ ∗ z that are properly convex and P r z ∈ΛΓ Γ-invariant, as in Example 3.7. However, if Γ is irreducible, then the following holds. Fact 4.7 ([B2, Prop. 3.1]). Let Γ be an irreducible discrete subgroup of PGL(V ) preserving a nonempty properly convex open subset Ω of P(V ). Then • Γ always contains a proximal element and the set Ωmax of Proposition 4.5.(3) is always properly convex (see Figure 3); it is a maximal Γ-invariant properly convex open subset of P(V ) containing Ω; • if moreover Γ is strongly irreducible (i.e. all finite-index subgroups of Γ are irreducible), then Ωmax is the unique maximal Γ-invariant properly convex open set in P(V ); it contains all other invariant properly convex open subsets; • in general, there is a smallest nonempty Γ-invariant convex open subset Ωmin of Ωmax, namely the interior of the convex hull of ΛΓ in Ωmax. In the irreducible case, using Lemma 4.1, we can describe the full orbital orb limit set ΛΩ (Γ) as follows. Proposition 4.8. Let Γ be an infinite irreducible discrete subgroup of PGL(V ) acting convex cocompactly on some nonempty properly convex open subset Ω 28 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Figure 3. The sets Ωmin (light gray) and Ωmax (dark gray) 2 for a convex cocompact subgroup of PO(2, 1). Here H is the disc bounded by the dashed circle. Note: Γ\Ccor (Γ) is Ωmax compact (equal to Γ\Ccor(Γ)) but Γ\Ccor (Γ) is not. H2 Ωmin of P(V ). Let Ωmin ⊂ Ω ⊂ Ωmax be nonempty properly convex open subsets of P(V ) with Ωmin minimal and Ωmax maximal for inclusion. Then orb ΛΩ (Γ) = ∂Ωmin ∩ ∂Ωmax, cor orb and the convex hull CΩ (Γ) of ΛΩ (Γ) in Ω is the convex hull of the proximal limit set ΛΓ in Ωmax, namely cor CΩ (Γ) = Ωmin ∩ Ωmax. cor Proof. By Lemma 4.1.(1), the set C := CΩ (Γ) is closed in Ω. Since the action of Γ on Ωmax is properly discontinuous (see Section 2.1), we have orb ∂iC = ΛΩ (Γ) ⊂ ∂Ωmax, hence C is also closed in Ωmax. The interior Int(C) is nonempty since Γ is irreducible, and Int(C) contains Ωmin. In fact Int(C) = Ωmin: indeed, if Ωmin were strictly smaller than Int(C), then the closure of Ωmin in C would be a nonempty strict closed subset of C on which Γ acts properly discontinuously and cocompactly, contradicting Lemma 4.1.(2). orb Thus C is the closure of Ωmin in Ωmax and ΛΩ (Γ) = Ωmin ∩ ∂Ωmax = ∂Ωmin ∩ ∂Ωmax. For an arbitrary irreducible discrete subgroup Γ of PGL(V ) preserving a cor nonempty properly convex open subset Ω of P(V ), the convex hull CΩ (Γ) of orb the full orbital limit set ΛΩ (Γ) may be larger than the convex hull Ωmin ∩ Ωmax of the proximal limit set ΛΓ in Ω. This happens for instance if Γ is naively convex cocompact in P(V ) (Deﬁnition 1.9) but not convex cocompact in P(V ). Examples of such behavior will be given in the forthcoming paper [DGK5].

4.4. The case of a connected proximal limit set. We make the following observation. CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 29

Lemma 4.9. Let Γ be an infinite discrete subgroup of PGL(V ) preserving a nonempty properly convex open subset Ω of P(V ). Suppose the proximal ∗ ∗ limit set ΛΓ of Γ in P(V ) (Definition 2.2) is connected. Then the set S ∗ (V ) ∗ ∗ z is a nonempty Γ-invariant convex open subset of (V ), not P r z ∈ΛΓ P necessarily properly convex, but containing all Γ-invariant properly convex open subsets of P(V ); it is equal to the set Ωmax of Proposition 4.5. Proof. By Proposition 4.5.(3), it is enough to observe the following basic ∗ ∗ fact: if L is a nonempty, closed, connected subset of P(V ), then P(V ) r S ∗ z∗∈L∗ z has at most one connected component, necessarily convex. To ∗ ∗ see this, regard the points of P(V ) = P((V ) ) as projective hyperplanes in ∗ S ∗ ∗ P(V ). The points of P(V )r z∗∈L∗ z are precisely the hyperplanes in P(V ) that miss L∗. Suppose there is such a hyperplane; then its complement in ∗ ∗ ∗ ∗ P(V ) is an affine chart A containing L . Let CH(L ) be the convex hull ∗ ∗ ∗ of L in the affine chart A ; it is a closed properly convex set. Since L is connected and closed, CH(L∗) is well-defined independent of the affine chart ∗ ∗ ∗ A : the convex hull of L in any affine chart containing L yields the same ∗ ∗ ∗ properly convex set CH(L ). Further, a hyperplane in P(V ) misses L if and only if it misses the convex hull CH(L∗), again because L∗ is closed S ∗ and connected. It follows that P(V ) r z∗∈L∗ z coincides with P(V ) r S ∗ z∗∈CH(L∗) z , which is a connected convex open set in P(V ). 4.5. Conical limit points. As a consequence of the proof of Lemma 4.1.(1), we observe that the dynamical behavior of orbits for general convex cocompact actions in P(V ) partially extends the dynamical behavior for classical convex cocompact subgroups of PO(d, 1). The contents of this subsection are not needed anywhere else in the paper. Definition 4.10. Let Γ be an infinite discrete subgroup of PGL(V ) and Ω a nonempty Γ-invariant properly convex open subset of P(V ). For y ∈ Ω and z ∈ ∂Ω, we say that z is a conical limit point of the orbit Γ · y if there exist a sequence (γm) ∈ ΓN and a projective line L in Ω such that γm · y → z and dΩ(γm · y, L) is bounded. We say that z is a conical limit point of Γ in ∂Ω if it is a conical limit point of the orbit Γ · y for some y ∈ Ω. Proposition 4.11. Let Γ be an infinite discrete subgroup of PGL(V ) and Ω a nonempty Γ-invariant properly convex open subset of P(V ). (1) If Γ acts cocompactly on some nonempty closed convex subset C of Ω, then the ideal boundary ∂iC consists entirely of conical limit points of Γ in ∂Ω. (2) In particular, if the action of Γ on Ω is convex cocompact (Defini- orb tion 1.11), then the full orbital limit set ΛΩ (Γ) consists entirely of conical limit points of Γ in ∂Ω.

Proof. (1) We proceed similarly to the proof of Lemma 4.1.(1). Let y ∈ ∂iC and let ym → y be a sequence in a ray [y0, y) in C. By cocompactness, there 0 −1 is a sequence (γm) in Γ such that ym := γm · ym stays in a compact subset 30 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

0 0 of C ⊂ Ω. Up to passing to a subsequence, we may assume ym → y ∈ C. We 0 0 have dΩ(ym, γm ·ym) → 0, and so (γm ·y )m remains within bounded distance (in fact is asymptotic to) [y0, y) and converges to y by Corollary 3.5. cor (2) Apply (1) to C = CΩ (Γ) and use Lemma 4.1.(1). When the boundary of Ω is strictly convex with C1 boundary, the property orb that ΛΩ (Γ) consist entirely of conical limit points implies that the action of Γ on Ω is convex cocompact [CM, Cor. 8.6]. The converse is false in general when ∂Ω is not strictly convex: indeed, in Example 3.9.(2), the action of Γ orb on Ω is not convex cocompact, but every point of ΛΩ (Γ) is conical. 5. Duality for convex cocompact actions In this section we study a notion of duality for properly convex sets which are not necessarily open. We prove Theorem 1.17.(A), and in fact a more precise version of it, namely Proposition 5.6. 5.1. The dual of a properly convex set. Given an open properly convex ∗ ∗ subset Ω ⊂ P(V ), there is a notion of dual convex set Ω ⊂ P(V ) which is very useful in the study of divisible convex sets: see Section 2.1. We generalize this notion here to properly convex sets with possibly nonempty nonideal boundary.

Deﬁnition 5.1. Let C be a properly convex subset of P(V ) with nonempty ∗ ∗ interior, but not necessarily open nor closed. The dual C ⊂ P(V ) of C is the set of projective hyperplanes in P(V ) that do not meet Int(C) ∪ ∂iC, i.e. those hyperplanes that meet C in a (possibly empty) subset of ∂nC. Remark 5.2. The interior of C∗ is the dual of the interior of C, in the ∗ ∗ usual sense for open convex subsets of P(V ). In fact C is convex in P(V ): 00 0 ∗ if hyperplanes H,H ,H ∈ C∗ ⊂ P(V ) are aligned in this order and if 0 ∗ 00 0 00 ∗ H,H ∈ C , then H ∩C ⊂ (H ∩H )∩C ⊂ ∂nC so H ∈ C . By construction, ∗ • Int(C ) is the set of projective hyperplanes in P(V ) that miss C; ∗ • ∂nC is the set of projective hyperplanes in P(V ) whose intersection with C is a nonempty subset of ∂nC; ∗ • ∂iC is the set of supporting projective hyperplanes of C at points of ∂iC (such a hyperplane misses ∂nC if C has bisaturated boundary). Lemma 5.3. Let C be a properly convex subset of P(V ), not necessarily open nor closed, but with bisaturated boundary. (1) The dual C∗ has bisaturated boundary. (2) The bidual (C∗)∗ coincides with C (after identifying (V ∗)∗ with V ). (3) The dual C∗ has a PET (Deﬁnition 1.16) if and only if C does. ∗ Proof. (1): By Remark 5.2, since C has bisaturated boundary, ∂nC (resp. ∗ ∂iC ) is the set of projective hyperplanes in P(V ) whose intersection with C ∗ is a nonempty subset of ∂nC (resp. ∂iC). In particular, a point of ∂nC and a ∗ point of ∂iC , seen as projective hyperplanes of P(V ), can only meet outside CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 31

∗ ∗ of C. This means exactly that a supporting hyperplane of C in P(V ) cannot ∗ ∗ meet both ∂nC and ∂iC . ∗ ∗ ∗ ∗ (2): By definition, (C ) is the set of hyperplanes of P(V ) missing Int(C )∪ ∗ ∗ ∂iC . Viewing hyperplanes of P(V ) as points of P(V ), by Remark 5.2 the ∗ ∗ set (C ) consists of those points of P(V ) not belonging to any hyperplane that misses C, or any supporting hyperplane at a point of ∂iC. Since C has bisaturated boundary, this is C r ∂iC, namely C. (3): By (1) and (2) it is enough to prove one implication. Suppose C has a PET contained in a two-dimensional projective plane P , i.e. C ∩ P = T is an open triangle. Let H1,H2,H3 be projective hyperplanes of P(V ) supporting C and containing the edges E1,E2,E3 ⊂ ∂iC of T . For 1 ≤ k ≤ 3, the supporting hyperplane Hk intersects ∂iC, hence lies in the ideal boundary ∗ ∗ ∂iC of the dual convex. Since C has bisaturated boundary by (1), the whole ∗ ∗ edge [Hk,Hk0 ] ⊂ C is contained in ∂iC . Hence H1,H2,H3 span a 2-plane ∗ ∗ ∗ Q ⊂ P(V ) whose intersection with C is a PET of C . 5.2. Proper and cocompact actions on the dual. The following is the key ingredient in Theorem 1.17.(A). Proposition 5.4. Let Γ be a discrete subgroup of PGL(V ) and C a Γ-invariant convex subset of P(V ) with bisaturated boundary. The action of Γ on C is properly discontinuous and cocompact if and only if the action of Γ on C∗ is. In order to prove Proposition 5.4, we assume n = dim(V ) ≥ 2 and first make some definitions. Consider a properly convex open subset Ω in P(V ). For any distinct points y, z ∈ P(V )r∂Ω, the line L through y and z intersects ∂Ω in two points a, b and we set δΩ(y, z) := max [a, y, z, b], [b, y, z, a] .

If y, z ∈ Ω, then δ(y, z) = exp(2dΩ(y, z)) > 1 where dΩ is the Hilbert metric on Ω (see Section 2.1). However, if y ∈ Ω and z ∈ P(V ) r Ω, then we have −1 ≤ δΩ(y, z) < 0. For any point y ∈ Ω and any projective hyperplane H ∈ Ω∗ (i.e. disjoint from Ω), we set

δΩ(y, H) := max δΩ(y, z). z∈H Then δ(y, H) ∈ [−1, 0) is close to 0 when H is “close” to ∂Ω as seen from y.

Lemma 5.5. Let Ω be a nonempty properly convex open subset of P(V ) = n ∗ P(R ). For any H ∈ Ω , there exists y ∈ Ω such that −1 δ (y, H) ≤ . Ω n − 1 ∗ n−1 Proof. Fix H ∈ Ω and consider an affine chart R of P(V ) for which H is at infinity, endowed with a Euclidean norm k · k. We take for y the center of mass of Ω in this affine chart with respect to the Lebesgue measure. It is enough to show that if a, b ∈ ∂Ω satisfy y ∈ [a, b], then ky − ak/kb − ak ≤ n−1 (n − 1)/n. Up to translation, we may assume a = 0 ∈ R . Let ` be 32 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

n−1 a linear form on R such that `(b) = 1 = supΩ `. Let h := `(y), so that 0 −1 ky −ak/kb−ak = h, and let Ω := R>0 ·(Ω∩` (h))∩{` < 1} (see Figure 4). The average value of ` on Ω0 for the Lebesgue measure is at least that on Ω,

b Ω0 Ω {` = 1} {` = h} y

a = 0

Figure 4. Illustration for the proof of Lemma 5.5 since by convexity Ω0 ∩ `−1(−∞, h] ⊂ Ω ∩ `−1(−∞, h], Ω0 ∩ `−1[h, 1) ⊃ Ω ∩ `−1[h, 1). The average value of ` on Ω is `(y) = h since y is the center of mass of Ω. The average value of ` on Ω0 is (n − 1)/n since Ω0 is a truncated open cone n−1 in R . Proof of Proposition 5.4. By Lemma 5.3, the set C∗ has bisaturated boundary and (C∗)∗ = C. Thus it is enough to prove that if the action on C is properly discontinuous and cocompact, then so is the action on C∗. Let us begin with properness. Recall from Lemma 3.6 that the set ∂iC is closed in P(V ). Let C0 be the convex hull of ∂iC in C. Note that any supporting hyperplane of C0 contains a point of ∂iC: otherwise it would be separated from ∂iC by a hyperplane, contradicting the deﬁnition of C0. By Lemma 4.3.(2), we have C0 ⊂ Ω := Int(C). Let C1 be the closed uniform 1-neighborhood of C0 in (Ω, dΩ). It is properly convex by [Bu, (18.12)], with nonempty interior, and ∂nC1 ∩ ∂nC = ∅. Taking the dual, we obtain ∗ ∗ that Int(C1) is a Γ-invariant properly convex open set containing C . In ∗ ∗ particular, the action of Γ on C ⊂ Int(C1) is properly discontinuous (see Section 2.1). Let us show that the action of Γ on C∗ is cocompact. Let Ω = Int(C) and Ω∗ = Int(C∗). Let D ⊂ C be a compact fundamental domain for the action of Γ on C. Consider the following subset of C∗, where n := dim(V ): ∗ [ ∗ −1 U = H ∈ Ω | δΩ(y, H) ≤ n−1 . y∈D∩Ω It follows from Lemma 5.5 that Γ ·U ∗ = Ω∗. We claim that U ∗ ⊂ C∗. To see ∗ ∗ this, suppose a sequence of elements Hm ∈ U converges to some H ∈ U ; let us show that H ∈ C∗. If H ∈ Ω∗ there is nothing to prove, so we may assume that H ∈ ∂Ω∗ = Fr(C∗) is a supporting hyperplane of C at a point −1 z ∈ Fr(C). For every m, let ym ∈ D ∩ Ω satisfy δΩ(ym,Hm) ≤ n−1 . Up to passing to a subsequence, we may assume ym → y ∈ D ⊂ C. Let zm ∈ Hm CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 33 such that zm → z. For every m, let am, bm ∈ ∂Ω be such that am, ym, bm, zm are aligned in this order. Then

−1 kbm − zmk ≥ δΩ(ym, zm) ≥ [bm, ym, zm, am] ≥ − , n − 1 kbm − ymk where k · k is a fixed Euclidean norm on an affine chart containing Ω. Since kbm − zmk → 0, we deduce kbm − ymk → 0, and so y = z ∈ C ∩ H ⊂ ∂nC. Since C has bisaturated boundary, we must have H ∈ C∗, by definition of C∗. Therefore U ∗ ⊂ C∗. Since U ∗ is compact and the action on C∗ is properly ∗ ∗ ∗ discontinuous, the fact that Γ ·U = Ω yields Γ · U ∗ = C . 5.3. Proof of Theorem 1.17.(A). We prove in fact the following more precise result, which implies Theorem 1.17.(A). Proposition 5.6. An infinite discrete subgroup Γ of PGL(V ) is convex co- ∗ compact in P(V ) if and only if it is convex cocompact in P(V ). In this case, there is a nonempty Γ-invariant properly convex open subset Ω of P(V ) such that the actions of Γ on Ω and on its dual Ω∗ are both convex cocompact, orb ∗ orb and such that Ω r ΛΩ (Γ) and Ω r ΛΩ∗ (Γ) both have bisaturated boundary. Proof. Suppose Γ is convex cocompact in P(V ). By the implication (1) ⇒ (2) of Theorem 1.20 (proved in Section 4.1), the group Γ acts properly discontinuously and cocompactly on a nonempty properly convex set C ⊂ P(V ) ∗ ∗ with bisaturated boundary. By Lemma 5.3.(1), the dual C ⊂ P(V ) has bisaturated boundary, and Γ acts properly discontinuously and cocompactly on C∗ by Proposition 5.4. By the implication (2) ⇒ (1) of Theorem 1.20 ∗ (proved in Section 4.2), the group Γ is convex cocompact in P(V ). Let Ω be the interior of C. By construction (see Remark 5.2), the dual Ω∗ of Ω is the interior of the dual C∗ of C. By Corollary 4.4, the actions of Γ on Ω ∗ orb and on its dual Ω are both convex cocompact, and we have ΩrΛΩ (Γ) = C ∗ orb ∗ and Ω r ΛΩ∗ (Γ) = C ; these convex sets have bisaturated boundary.

Note that in the example of Figure 3, the action of Γ on Ω := Ωmax is ∗ convex cocompact while the action on its dual Ω = Ωmin is not.

6. Segments in the full orbital limit set In this section we establish the equivalences (ii) ⇔ (iii) ⇔ (iv) ⇔ (v) in Theorem 1.15. For this it will be helpful to introduce two intermediate conditions, weaker than (iv) but stronger than (iii): (iii)’ Γ is convex cocompact in P(V ) and for some nonempty properly orb convex open set Ω on which Γ acts convex cocompactly, ΛΩ (Γ) does not contain a nontrivial projective line segment; (iv)’ Γ is convex cocompact in P(V ) and for any nonempty properly con- cor vex open set Ω on which Γ acts convex cocompactly, CΩ (Γ) does not contain a PET. 34 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

The implications (iii) ⇒ (iii)’, (iv)’ ⇒ (iv), (iii) ⇒ (iv)’, and (iii)’ ⇒ (iv) are trivial. The implication (ii) ⇒ (iv)’ holds by Lemma 6.1 below. In Section 6.2 we simultaneously prove (iv)’ ⇒ (iii) and (iv) ⇒ (iii)’. In Section 6.3 we prove (iii)’ ⇒ (ii). In Section 6.4 we prove (v) ⇒ (iii)’, and in Section 6.5 we prove (iii)’ ⇒ (v). We include the following diagram of implications for the reader’s convenience:

(iii): (iii)’: for any cc Ω, +3 for some cc Ω, orb orb ΛΩ 6⊃ segment ΛΩ 6⊃ segment FN PX W_ § 6.5

§ 6.3 ! yÑ § 6.4 (v): (ii): ∃Ω ⊃ C § 6.2 Γ cc and § 6.2 cocompact Γ hyperbolic ∂iC 6⊃ segment

yÑ (iv)’: (iv): for any cc Ω, +3 for some cc Ω, cor cor CΩ 6⊃ PET CΩ 6⊃ PET 6.1. PETs obstruct hyperbolicity. We start with an elementary remark. Lemma 6.1. Let Γ be an inﬁnite discrete subgroup of PGL(V ) preserving a properly convex open subset Ω and acting cocompactly on a closed convex subset C of Ω. If C contains a PET (Deﬁnition 1.16), then Γ is not word hyperbolic.

Proof. A PET in Ω is totally geodesic for the Hilbert metric dΩ and is quasi- isometric to the Euclidean plane. Thus, if C contains a PET, then (C, dΩ) cannot be Gromov hyperbolic, hence Γ cannot be word hyperbolic by the Švarc–Milnor lemma. 6.2. Segments in the full orbital limit set yield PETs in the convex core. The implications (iv)’ ⇒ (iii) and (iv) ⇒ (iii)’ in Theorem 1.15 will be a consequence of the following lemma, which is similar to [B6, Prop. 3.8.(a)] but without the divisibility assumption nor the restriction to dimension 3; our proof is diﬀerent, close to [B3]. Lemma 6.2. Let Γ be an inﬁnite discrete subgroup of PGL(V ). Let Ω be a nonempty Γ-invariant properly convex open subset of P(V ). Suppose Γ acts cocompactly on some nonempty closed convex subset C of Ω. If ∂iC contains a nontrivial segment which is inextendable in ∂Ω, then C contains a PET.

Proof. The ideal boundary ∂iC is closed in P(V ) by Lemma 3.6. Suppose ∂iC contains a nontrivial segment [a, b] which is inextendable in ∂Ω. Let CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 35 c ∈ C and consider a sequence of points ym ∈ C lying inside the triangle with vertices a, b, c and converging to a point in y ∈ (a, b). We claim that the dΩ-distance from ym to either projective interval (a, c] or (b, c] tends to infinity with m. Indeed, consider a sequence (zm)m of points of (a, c] converging to z ∈ [a, c] and let us check that dΩ(ym, zm) → +∞ (the proof for (b, c] is the same). If z ∈ (a, c], then z ∈ C and so dΩ(ym, zm) → +∞ by properness of the Hilbert metric. Otherwise z = a. In that case, for 0 0 0 0 each m, consider ym, zm ∈ ∂Ω such that ym, ym, zm, zm are aligned in that 0 0 0 0 order. Up to taking a subsequence, we may assume ym → y and zm → z for some y0, z0 ∈ ∂Ω, with y0, y, a, z0 aligned in that order. By inextendability of 0 [a, b] in ∂Ω, we must have z = a = z , hence dΩ(ym, zm) → +∞ in this case as well, proving the claim. Since the action of Γ on C is cocompact, there is a sequence (γm) ∈ ΓN such that γm · ym remains in a fixed compact subset of C. Up to passing to a subsequence, we may assume that (γm · ym)m converges to some y∞ ∈ C, and (γm · a)m and (γm · b)m and (γm · c)m converge respectively to some a∞, b∞, c∞ ∈ ∂iC, with [a∞, b∞] ⊂ ∂iC since ∂iC is closed (Lemma 4.1.(1)). The triangle with vertices a∞, b∞, c∞ is nondegenerate since it contains y∞ ∈ C. Further, y∞ is infinitely far (for the Hilbert metric on Ω) from the edges [a∞, c∞] and [b∞, c∞], and so these edges are fully contained in ∂iC. Thus the triangle with vertices a∞, b∞, c∞ is a PET of C. Simultaneous proof of (iv)’ ⇒ (iii) and (iv) ⇒ (iii)’. We prove the contra- positive. Suppose Γ ⊂ PGL(V ) acts convex cocompactly on the properly orb convex open set Ω ⊂ P(V ) and ΛΩ (Γ) contains a nontrivial segment. By cor orb Lemma 4.1.(1), the set ∂iCΩ (Γ) is equal to ΛΩ (Γ) and closed in P(V ), and it contains the open stratum of ∂Ω at any interior point of that segment; orb in particular, ΛΩ (Γ) contains a nontrivial segment which is inextendable in cor cor ∂Ω. By Lemma 6.2 with C = CΩ (Γ), the set CΩ (Γ) contains a PET. 6.3. Word hyperbolicity of the group in the absence of segments. In this section we prove the implication (iii)’ ⇒ (ii) in Theorem 1.15. We proceed exactly as in [DGK3, § 4.3], with arguments inspired from [B3]. It is cor sufficient to apply the following general result to C = CΩ . Recall that any geodesic ray of (C, d) has a well-defined endpoint in ∂iC (see [FK, Th. 3] or [DGK3, Lem. 2.6.(1)]). Lemma 6.3. Let Γ be a discrete subgroup of PGL(V ). Let Ω be a nonempty Γ-invariant properly convex open subset of P(V ). Suppose Γ acts cocompactly on some nonempty closed convex subset C of Ω such that ∂iC does not contain any nontrivial projective line segment. Then

(1) there exists R > 0 such that any geodesic ray of (C, dΩ) lies at Haus- dorﬀ distance ≤ R from the projective interval with the same endpoints; (2) the metric space (C, dΩ) is Gromov hyperbolic with Gromov boundary Γ-equivariantly homeomorphic to ∂iC; 36 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

(3) the group Γ is word hyperbolic and any orbit map Γ → (C, dΩ) is a quasi-isometric embedding which extends to a Γ-equivariant homeomorphism ξ : ∂∞Γ → ∂iC. cor (4) if C contains CΩ (Γ) (hence Γ acts convex cocompactly on Ω), then any orbit map Γ → (Ω, dΩ) is a quasi-isometric embedding and ex- orb tends to a Γ-equivariant homeomorphism ξ : ∂∞Γ → ΛΩ (Γ) = ∂iC which is independent of the orbit.

As usual, we denote by dΩ the Hilbert metric on Ω.

Proof. (1) Suppose by contradiction that for any m ∈ N there is a geodesic ray Gm of (C, dΩ) with endpoints am ∈ C and bm ∈ ∂iC and a point ym ∈ C on that geodesic which lies at distance ≥ m from the projective interval [am, bm). By cocompactness of the action of Γ on C, for any m ∈ N there exists γm ∈ Γ such that γm · ym belongs to a fixed compact set of C. Up to taking a subsequence, (γm · ym)m converges to some y∞ ∈ C, and (γm · am)m and (γm · bm)m converge respectively to some a∞ ∈ C and b∞ ∈ ∂iC. Since the distance from ym to [am, bm) goes to infinity, we have [a∞, b∞] ⊂ ∂iC, hence a∞ = b∞ since ∂iC does not contain a segment. Therefore, up to extracting, the geodesics Gm converge to a biinfinite geodesic of (Ω, dΩ) with both endpoints equal. But such a geodesic does not exist (see [FK, Th. 3] or [DGK3, Lem. 2.6.(2)]): contradiction. (2) Suppose by contradiction that triangles of (C, dΩ) are not uniformly thin. By (1), triangles of (C, dΩ) whose sides are projective line segments are not uniformly thin: namely, there exist am, bm, cm ∈ C and ym ∈ [am, bm] such that

(6.1) dΩ(ym, [am, cm] ∪ [cm, bm]) −→ +∞. m→+∞

By cocompactness, for any m there exists γm ∈ Γ such that γm · ym belongs to a fixed compact set of C. Up to taking a subsequence, (γm ·ym)m converges to some y∞ ∈ C, and (γm · am)m and (γm · bm)m and (γm · cm)m converge respectively to some a∞, b∞, c∞ ∈ C. By (6.1) we have [a∞, c∞] ∪ [c∞, b∞] ⊂ ∂iC, hence a∞ = b∞ = c∞ since ∂iC does not contain any nontrivial projective line segment. This contradicts the fact that y∞ ∈ (a∞, b∞). Therefore (C, dΩ) is Gromov hyperbolic. Fix a basepoint y ∈ C. The Gromov boundary of (C, dΩ) is the set of equivalence classes of infinite geodesic rays in C starting at y, for the equivalence relation “to remain at bounded distance for dΩ”. Consider the Γ-equivariant continuous map ϕ from ∂iC to this Gromov boundary sending z ∈ ∂iC to the class of the geodesic ray from y to z. This map is clearly surjective, since any infinite geodesic ray in C terminates at the ideal boundary ∂iC. Moreover, it is injective, since the non-existence of line segments in ∂iC means that no two points of ∂iC lie in a common face of ∂Ω, hence the Hilbert distance between rays going out to two different points of ∂iC goes to infinity. CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 37

(3) The group Γ acts properly discontinuously and cocompactly, by isometries, on the metric space (C, dΩ), which is Gromov hyperbolic with Gromov boundary ∂iC by (2) above. We apply the Švarc–Milnor lemma. cor (4) Suppose C contains CΩ (Γ). Consider a point y ∈ Ω. It lies in a uniform neighborhood Cy of C in (Ω, dΩ), on which the action of Γ is also orb cocompact. By Lemma 4.1.(1)–(2), we have ∂iCy = ∂iC = ΛΩ (Γ), and this set contains no nontrivial segment by assumption. By (3), the orbit map Γ → (Cy, dΩ) associated with y is a quasi-isometry which extends to a Γ-equivariant homeomorphism ∂∞Γ → ∂iCy = ∂iC, and this extension is clearly independent of y.

6.4. Proof of (v) ⇒ (iii)’ in Theorem 1.15. It is suﬃcient to prove the orb following lemma: then Γ is convex cocompact in P(V ) and ∂iC = ΛΩ (Γ) by Lemma 4.1.(1).

Lemma 6.4. Let Γ be an inﬁnite discrete subgroup of PGL(V ) preserving a properly convex open subset Ω of P(V ) and acting cocompactly on some closed convex subset C of Ω with nonempty interior. If ∂iC does not contain cor any nontrivial segment, then C contains CΩ (Γ). In particular, if C is as in Lemma 6.3 and has nonempty interior, then the conclusion of Lemma 6.3.(4) holds. The proof of Lemma 6.4 is similar to [DGK3, Lem. 4.3].

Proof. Suppose that ∂iC does not contain any nontrivial segment. It is suf- orb ﬁcient to prove that ΛΩ (Γ) ⊂ ∂iC. Suppose by contradiction that there orb exists a point z∞ in ΛΩ (Γ) r ∂iC. We can write z∞ = limm γm · z for some (γm) ∈ ΓN and z ∈ Ω r C. Let y ∈ Int(C). The segment [y, z] contains a point w ∈ ∂nC in its interior. Up to passing to a subsequence, we may assume that (γm · y)m∈N and (γm · w)m∈N converge respectively to y∞ and w∞, with [y∞, w∞] ⊂ ∂iC. Since ∂iC does not contain any nontrivial segment, we have y∞ = w∞. Let (a, b) be the maximal interval of Ω containing a, y, w, z, b in that order, used to compute the Hilbert distances between the points y, w, z. Since dΩ(γm ·y, γm ·w) = dΩ(y, w) > 0 and y∞ = w∞, it follows that at least one of a∞ = limm γm ·am or b∞ = limm γm ·bm is equal to y∞ = w∞. We cannot have b∞ = y∞, since y∞ 6= z∞ and y∞, z∞, b∞ are aligned in this order. Therefore a∞ = y∞. But then z∞ = y∞ since dΩ(γm · y, γm · z) = dΩ(y, z) is bounded: contradiction.

6.5. Proof of (iii)’ ⇒ (v) in Theorem 1.15. Suppose that Γ acts convex cocompactly on some nonempty properly convex open subset Ω of P(V ) orb and that ΛΩ (Γ) does not contain any nontrivial projective line segment. The group Γ acts cocompactly on the closed uniform 1-neighborhood C of cor CΩ (Γ) in (Ω, dΩ), which is properly convex with nonempty interior. By orb Lemma 4.1.(1), we have ∂iC = ΛΩ (Γ). 38 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

7. Convex cocompactness and no segment implies P1-Anosov In this section we continue with the proof of Theorem 1.15. We have already established the equivalences (ii) ⇔ (iii) ⇔ (iv) ⇔ (v) in Section 6. On the other hand, the implication (i) ⇒ (iii) is trivial. We now prove the implication (ii) ⇒ (vi) in Theorem 1.15. By the above, this yields the implication (i) ⇒ (vi) in Theorem 1.15, which is also the implication (1) ⇒ (2) in Theorem 1.4. We build on Lemma 6.3.(3).

7.1. Compatible, transverse, dynamics-preserving boundary maps. Let Γ be an inﬁnite discrete subgroup of PGL(V ). Suppose Γ is word hyperbolic and convex cocompact in P(V ). By Proposition 5.6, there is a nonempty Γ-invariant properly convex open subset Ω of P(V ) such that the actions of Γ on Ω and on its dual Ω∗ are both convex cocompact. Our goal is to show that the natural inclusion Γ ,→ PGL(V ) is P1-Anosov. By the implication (ii) ⇒ (iii) in Theorem 1.15 (which we have proved in Section 6) and Theorem 1.17.(A) (which we have proved in Section 5), the orb orb ∗ full orbital limit sets ΛΩ (Γ) ⊂ P(V ) and ΛΩ∗ (Γ) ⊂ P(V ) do not contain any cor cor ∗ nontrivial projective line segment. Let CΩ ⊂ P(V ) (resp. CΩ∗ ⊂ P(V )) be orb orb ∗ the convex hull of ΛΩ (Γ) in Ω (resp. of ΛΩ∗ (Γ) in Ω ). By Lemma 6.3.(3), cor cor any orbit map Γ → (CΩ , dΩ) (resp. Γ → (CΩ∗ , dΩ∗ )) is a quasi-isometry which extends to a Γ-equivariant homeomorphism orb ξ : ∂∞Γ −→ ΛΩ (Γ) ⊂ P(V ) (resp. ∗ orb ∗ ξ : ∂∞Γ −→ ΛΩ∗ (Γ) ⊂ P(V )). ∗ We see P(V ) as the space of projective hyperplanes of P(V ). Lemma 7.1. The boundary maps ξ and ξ∗ are compatible, i.e. for any η ∈ ∗ ∗ ∂∞Γ we have ξ(η) ∈ ξ (η); more precisely, ξ (η) is a supporting hyperplane of Ω at ξ(η).

Proof. Let (γm)m∈N be a quasi-geodesic ray in Γ with limit η ∈ ∂∞Γ. For ∗ ∗ any y ∈ Ω and any H ∈ Ω , we have γm · y → ξ(η) and γm · H → ξ (η). Lift y to a vector x ∈ V and lift H to a linear form ϕ ∈ V ∗. Lift the ± sequence γm to a sequence γˆm ∈ SL (V ). By Lemma 3.2, the sequences ∗ (ˆγm · x)m∈N and (ˆγm · ϕ)m∈N go to inﬁnity in V and V respectively. Since ∗ (ˆγm · ϕ)(ˆγm · x) = ϕ(x) is independent of m, we obtain ξ(η) ∈ ξ (η) by passing to the limit. Since ξ∗(η) belongs to ∂Ω∗, it is a supporting hyperplane of Ω. ∗ 0 Lemma 7.2. The maps ξ and ξ are transverse, i.e. for any η 6= η in ∂∞Γ we have ξ(η) ∈/ ξ∗(η0).

0 ∗ 0 Proof. Consider η, η ∈ ∂∞Γ such that ξ(η) ∈ ξ (η ). Let us check that 0 orb cor η = η . By Lemma 4.1.(1), we have ΛΩ (Γ) = ∂iCΩ (Γ), and so the projective line segment [ξ(η), ξ(η0)], contained in the supporting hyperplane ξ∗(η0), is CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 39

orb orb contained in ΛΩ (Γ). Since ΛΩ (Γ) contains no nontrivial projective line 0 segment by condition (iii) of Theorem 1.15, we deduce η = η . + − For any γ ∈ Γ of infinite order, we denote by ηγ (resp. ηγ ) the attracting (resp. repelling) fixed point of γ in ∂∞Γ. Lemma 7.3. The maps ξ and ξ∗ are dynamics-preserving. Proof. We only prove it for ξ; the argument is similar for ξ∗. We fix a norm k·kV on V . Let γ ∈ Γ be an element of infinite order; we lift it to an element ± γˆ ∈ SL (V ) that preserves a properly convex cone of V r {0} lifting Ω. Let + + − L be the line of V corresponding to ξ(ηγ ) and H the hyperplane of V ∗ − + − corresponding to ξ (ηγ ). By transversality, we have V = L ⊕ H , and this decomposition is preserved by γˆ. Let [x] ∈ Ω and write x = `+ + h− with `+ ∈ L+ and h− ∈ H−. Since Ω is open, we may choose x so that `+ 6= 0 − m − m − and h satisfies kγˆ · h kV ≥ δt kh kV > 0 for all m ∈ N, where δ > 0 and where t > 0 is the spectral radius of the restriction of γˆ to H−. On the other hand, γˆm · `+ = sm`+ where s is the eigenvalue of γˆ on L+. By m + Lemma 6.3.(4), we have γˆ · [x] → ξ(ηγ ) as m → +∞, hence m − kγˆ · h kV m + −→ 0. kγˆ · ` kV m→+∞ + Necessarily s > t, and so ξ(ηγ ) is an attracting fixed point for the action of γ on P(V ). As an immediate consequence of Lemma 6.3.(3) and Lemma 7.3, we obtain the following. Corollary 7.4. For any infinite-order element γ ∈ Γ, the element ρ(γ) ∈ orb PGL(V ) is proximal in P(V ), and the full orbital limit set ΛΩ (Γ) is the closure of the set of attracting fixed points of such elements.

7.2. The natural inclusion Γ ,→ PGL(V ) is P1-Anosov.

Lemma 7.5. We have (µ1 − µ2)(γ) → +∞ as γ → ∞ in Γ.

Proof. Consider a sequence (γm) ∈ ΓN going to inﬁnity in Γ. Up to extracting we can assume that there exists η ∈ ∂∞Γ such that γm → η. Then γm · y → ξ(η) for all y ∈ Ω by Lemma 6.3.(3). For any m we can write γm = 0 + kmamkm ∈ K(exp a )K where am = diag(ai,m)1≤i≤n with ai,m ≥ ai+1,m (see Section 2.2). Up to extracting, we may assume that (km)m∈N converges 0 0 to some k ∈ K and (km)m∈N to some k ∈ K. Since Ω is open, we can ﬁnd points y, z of Ω lifting respectively to w, x ∈ V with n n 0 X 0 X k · w = siei and k · x = tiei i=1 i=1 such that s1 = t1 but s2 6= t2. For any m, let us write n n 0 X 0 X km · w = si,mei and km · x = ti,mei i=1 i=1 40 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL where si,m → si and ti,m → ti. Then " n # " n # X ai,m X ai,m a k0 ·y = s e + s e , a k0 ·z = t e + t e . m m 1,m 1 a i,m i m m 1,m 1 a i,m i i=2 1,m i=2 1,m

The sequences (γm ·y)m∈N and (γm ·z)m∈N both converge to ξ(η). Hence the 0 0 sequences (amkm ·y)m∈N and (amkm ·z)m∈N both converge to the same point −1 k · ξ(η) ∈ P(V ). Since s1 = t1 and s2 6= t2, we must have a2,m/a1,m → 0, i.e. (µ1 − µ2)(γm) → +∞.

By Fact 2.5, the natural inclusion Γ ,→ PGL(V ) is P1-Anosov. This completes the proof of the implication (ii) ⇒ (vi) of Theorem 1.15.

8. P1-Anosov implies convex cocompactness In this section we prove the implication (vi) ⇒ (ii) in Theorem 1.15. Let Γ be an infinite discrete subgroup of PGL(V ). Suppose that Γ is word hyperbolic, that the inclusion Γ ,→ PGL(V ) is P1-Anosov, and that Γ preserves some nonempty properly convex open subset O of P(V ). We wish to prove that Γ acts convex cocompactly on some nonempty properly convex open subset Ω of P(V ). We note that Ω cannot always be taken to be equal to O: see Remark 8.8 below. Recall that, by Proposition 4.5, when Γ preserves a nonempty properly ∗ ∗ convex open set O, the proximal limit set ΛΓ of Γ in P(V ) always lifts to a ∗ ∗ cone ΛeΓ of V r {0} such that the convex open set ∗ Ωmax := P({x ∈ V | `(x) > 0 ∀` ∈ ΛeΓ}) contains O and is Γ-invariant. Thus it is sufficient to prove the following. Proposition 8.1. Let Γ be an infinite discrete subgroup of PGL(V ) which is word hyperbolic, such that the natural inclusion Γ ,→ PGL(V ) is P1-Anosov, ∗ ∗ with boundary maps ξ : ∂∞Γ → P(V ) and ξ : ∂∞Γ → P(V ). Suppose that ∗ ∗ ∗ ∗ the proximal limit set ΛΓ = ξ (∂∞Γ) lifts to a cone ΛeΓ of V r {0} such that the convex open set ∗ Ωmax := P({x ∈ V | `(x) > 0 ∀` ∈ ΛeΓ}) is nonempty and Γ-invariant. Then Γ acts convex cocompactly (Defini- tion 1.11) on some nonempty properly convex open set Ω ⊂ Ωmax (which can be taken to be Ωmax if Ωmax is properly convex). Moreover, the full orb orbital limit set ΛΩ (Γ) for any such Ω is equal to the proximal limit set ΛΓ = ξ(∂∞Γ). The rest of the section is devoted to the proof of Proposition 8.1. 8.1. Convergence for Anosov representations. We first make the following general observation. Lemma 8.2. Let Γ be an infinite discrete subgroup of PGL(V ) which is word hyperbolic, such that the natural inclusion Γ ,→ PGL(V ) is P1-Anosov, with CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 41

∗ ∗ boundary maps ξ : ∂∞Γ → P(V ) and ξ : ∂∞Γ → P(V ). Let (γm)m∈N be a −1 sequence of elements of Γ converging to some η ∈ ∂∞Γ, such that (γm )m∈N 0 converges to some η ∈ ∂∞Γ. Then ∗ 0 (1) for any y ∈ P(V ) with y∈ / ξ (η ) we have γm · y → ξ(η); ∗ ∗ 0 ∗ ∗ ∗ (2) for any y ∈ P(V ) with ξ(η ) ∈/ y we have γm · y → ξ (η). Moreover, convergence is uniform for y, y∗ ranging over compact sets. Proof. The two statements are dual to each other, so we only need to prove (1). 0 + For any m ∈ N, we write γm = kmamkm ∈ K exp(a )K (see Section 2.2). 0 Up to extracting, (km) converges to some k ∈ K, and (km) converges to some k0 ∈ K. Note that −1 0 −1 −1 −1 + γm = (km w0)(w0am w0)(w0km ) ∈ KA K, n where w0 ∈ PGL(V ) ' PGL(R ) is the image of the permutation matrix exchanging ei and en+1−i. Therefore, by Fact 2.6, we have ∗ 0 0−1 ξ (η ) = k w0 · P(span(e1, . . . , en−1)). ∗ 0 In particular, for any y ∈ P(V ) with y∈ / ξ (η ) we have 0 k · y∈ / w0 · P(span(e1, . . . , en−1)) = P(span(e2, . . . , en)).

On the other hand, writing am = diag(am,i)1≤i≤n, we have am,1/am,2 = (µ1−µ2)(γm) 0 e → +∞ by Fact 2.5. Therefore amkm · y → [e1], and so γm · y = 0 kmamkm · y → k · [e1]. By Fact 2.6, we have k · [e1] = ξ(η). This proves (1). 0 Pn Uniformity follows from the fact k · y = [e1 + i=2 uiei] with bounded ui, ∗ 0 when y ranges over a compact set disjoint from the hyperplane ξ (η ). Corollary 8.3. In the setting of Lemma 8.2, for any nonempty Γ-invariant orb properly convex open subset Ω of P(V ), the full orbital limit set ΛΩ (Γ) is equal to the proximal limit set ΛΓ = ξ(∂∞Γ). ∗ ∗ ∗ ∗ 0 Proof. By Proposition 4.5, we have ΛΓ = ξ (∂∞Γ) ⊂ ∂Ω , hence y∈ / ξ (η ) 0 for all y ∈ Ω and η ∈ ∂∞Γ. We then apply Lemma 8.2.(1) to get both orb orb ΛΓ ⊂ ΛΩ (Γ) and ΛΩ (Γ) ⊂ ΛΓ. 8.2. Proof of Proposition 8.1. Let Γ be an inﬁnite discrete subgroup of PGL(V ). Suppose that Γ is word hyperbolic, that the natural inclusion ∗ ∗ Γ ,→ PGL(V ) is P1-Anosov, and that the proximal limit set ΛΓ = ξ (∂∞Γ) ∗ ∗ lifts to a cone ΛeΓ of V r {0} such that the convex open set ∗ Ωmax := P({x ∈ V | `(x) > 0 ∀` ∈ ΛeΓ}) is nonempty and Γ-invariant (see Figure 3). Let C0 be the convex hull of ΛΓ in Ωmax; it is a nonempty closed convex subset of Ωmax. In fact, C0 is properly convex because any point of ∂Ωmax that is contained in a full projective line of Ωmax must also be contained in ∗ ∗ every hyperplane of ΛΓ hence, by transversality of ΛΓ and ΛΓ, it must not be contained in ΛΓ.

Lemma 8.4. We have ∂iC0 = ΛΓ. 42 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Proof. By construction, ∂iC0 = C0 ∩ ∂Ωmax ⊃ ΛΓ. Suppose a point y ∈ C0 ∗ lies in ∂Ωmax. Then y lies in a hyperplane ξ (η) for some η ∈ ∂∞Γ. Then any minimal subset S of ΛΓ for which y lies in the convex hull of S must also be contained in ξ∗(η). By the transversality of ξ and ξ∗, this implies y ∈ ΛΓ.

The set Ωmax is convex, but it may fail to be properly convex when Γ is not irreducible (see Example 10.11). However, we observe the following. Lemma 8.5. There exists a Γ-invariant properly convex open subset Ω containing C0.

Proof. If Ωmax is properly convex, then we take Ω = Ωmax. Suppose that Ωmax fails to be properly convex. Choose projective hyperplanes H0,...,Hn bounding an open simplex ∆ containing C0. Deﬁne V := Ωmax ∩ ∆ and \ Ω := γ ·V. γ∈Γ

Then Ω contains C0 and is properly convex. We must show Ω is open. Suppose not and let z ∈ ∂nΩ. Then there is a sequence (γm) ∈ ΓN and i ∈ {1, . . . , `} such that γm · Hi converges to a hyperplane H∞ containing z. ∗ By Lemma 8.2.(2), the hyperplane H∞ lies in ΛΓ, contradicting the fact that ∗ the hyperplanes of ΛΓ are disjoint from Ωmax. Lemma 8.6. For any Γ-invariant properly convex open subset Ω of P(V ) cor containing C0, we have CΩ (Γ) = C0. orb cor Proof. By Corollary 8.3, we have ΛΩ (Γ) = ΛΓ, hence CΩ (Γ) is the convex cor hull of ΛΓ in Ω, i.e. CΩ (Γ) = C0 ∩Ω. But C0 is contained in Ω by construction cor (see Lemma 8.5), hence CΩ (Γ) = C0. In order to conclude the proof of Proposition 8.1, it only remains to check the following.

Lemma 8.7. The action of Γ on C0 is cocompact. Proof. By [KLPa, Th. 1.7] (see also [GGKW, Rem. 5.15]), the action of Γ on P(V ) at any point z ∈ ΛΓ is expanding: there exist an element γ ∈ Γ, a neighborhood U of z in P(V ), and a constant c > 1 such that γ is c-expanding on U for the metric 0 0 dP([x], [x ]) := | sin ](x, x )| on P(V ). We now use a version of the argument of [KLPa, Prop. 2.5], inspired by Sullivan’s dynamical characterization [Su] of convex cocompactness in the real hyperbolic space. (The argument in [KLPa] is a little more technical because it deals with bundles, whereas we work directly in P(V ).) Suppose by contradiction that the action of Γ on C0 is not cocompact, and let (εm)m∈N be a sequence of positive reals converging to 0. For any m, the set Km := {z ∈ C0 | dP(z, ΛΓ) ≥ εm} is compact (Lemma 8.4), hence there exists a Γ-orbit contained in C0 r Km. By proper discontinuity of the action CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 43 on C0, the supremum of dP(·, ΛΓ) on this orbit is achieved at some point zm ∈ C0, and by construction 0 < dP(zm, ΛΓ) ≤ εm. Then, for all γ ∈ Γ,

dP(γ · zm, ΛΓ) ≤ dP(zm, ΛΓ).

Up to extracting, we may assume that (zm)m∈N converges to some z ∈ ΛΓ. Consider an element γ ∈ Γ, a neighborhood U of z in P(V ), and a constant 0 c > 1 such that γ is c-expanding on U. For any m ∈ N, there exists zm ∈ ΛΓ 0 such that dP(γ · zm, ΛΓ) = dP(γ · zm, γ · zm). For large enough m we have 0 zm, zm ∈ U, and so 0 dP(γ · zm, ΛΓ) ≥ c dP(zm, zm) ≥ c dP(zm, ΛΓ) ≥ c dP(γ · zm, ΛΓ) > 0. This is impossible since c > 1. This shows that the word hyperbolic group Γ acts convex cocompactly on any Γ-invariant properly convex open subset Ω of P(V ) containing C0, hence condition (ii) of Theorem 1.15 is satisﬁed.

8.3. Two remarks on Proposition 8.1 and its proof. Remark 8.8. In the proof of Proposition 8.1 it is important, in order to obtain convex cocompactness, to consider only properly convex open sets Ω that contain C0. For instance, here would be two bad choices for Ω:

(1) if we took Ω to be the interior Int(C0) of C0, then Γ would not act cor convex cocompactly on Ω: we would have CΩ (Γ) = Ω = Int(C0), which does not have compact quotient by Γ; (2) if we took Ω such that C0 ⊂ ∂Ω, then Γ would not act convex cocom- cor pactly on Ω, as CΩ = ∅. Here is a concrete example for Remark 8.8.(2): suppose Γ is a convex cocompact subgroup (in the classical sense) of PO(d − 1, 1) ⊂ PO(d, 1) ⊂ d+1 PGL(R ). Then the set ΛΓ = ∂iC0 is contained in the equatorial sphere d−1 d d+1 ∂H of ∂H ⊂ P(R ). If we took Ω to be a Γ-invariant hyperbolic d half-space of H , then we would have C0 ⊂ ∂Ω. Remark 8.9. Suppose that Γ is word hyperbolic, that the natural inclusion Γ ,→ G = PGL(V ) is P1-Anosov, and that Γ preserves a nonempty properly convex open subset of P(V ). If the set Ωmax above is properly convex (which is always the case e.g. if Γ is irreducible), then we may take Ω = Ωmax in Lemmas 8.5 and 8.6, and so Γ acts convex cocompactly on Ω = Ωmax. orb orb However, if ∂Ω 6= ΛΩ (Γ), then the convex set Ω r ΛΩ (Γ) does not have bisaturated boundary, and it is possible to show that the action of Γ on orb Ω r ΛΩ (Γ) is not properly discontinuous. 8.4. The case of groups with connected boundary. Here is an immediate consequence of Propositions 4.5 and 8.1 and Lemma 4.9. Corollary 8.10. Let Γ be an inﬁnite discrete subgroup of PGL(V ) which is word hyperbolic with connected boundary ∂∞Γ, which preserves a properly 44 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL convex open subset of P(V ), and such that the natural inclusion Γ ,→ PGL(V ) is P1-Anosov. Then the set [ ∗ Ωmax := P(V ) r z ∗ ∗ z ∈ΛΓ is a nonempty Γ-invariant convex open subset of P(V ), containing all other such sets. If Ωmax is properly convex (e.g. if Γ is irreducible), then Γ acts convex cocompactly on Ωmax.

9. Smoothing out the nonideal boundary We now make the connection with strong projective convex cocompactness and prove the remaining implications of Theorems 1.15 and 1.20. Concerning Theorem 1.15, the equivalences (ii) ⇔ (iii) ⇔ (iv) ⇔ (v) have been proved in Section 6, the implication (ii) ⇒ (vi) in Section 7, and the implication (vi) ⇒ (ii) in Section 8. The implication (i) ⇒ (iii) is trivial. We shall prove (iii) ⇒ (i) in Section 9.2, which will complete the proof of Theorem 1.15. It will also complete the proof of Theorem 1.4, since the implication (2) ⇒ (1) of Theorem 1.4 is the implication (vi) ⇒ (i) of Theorem 1.15. Concerning Theorem 1.20, the equivalence (1) ⇔ (2) has been proved in Section 4. The implication (4) ⇒ (3) is immediate. We shall prove the implication (1) ⇒ (4) in Section 9.2. The implication (3) ⇒ (2) is an immediate consequence of Lemma 3.6 and of the following lemma, and completes the proof of Theorem 1.20.

Lemma 9.1. Let Cstrict be a nonempty convex subset of P(V ) with strictly convex nonideal boundary. If ∂iCstrict is closed in P(V ), then Cstrict has bisaturated boundary.

Proof. If ∂iCstrict is closed in P(V ), then a supporting hyperplane H of Cstrict at a point y ∈ ∂nCstrict cannot meet ∂iCstrict, otherwise it would need to contain a nontrivial segment of ∂nCstrict, which does not exist since Cstrict has strictly convex nonideal boundary. This shows that Cstrict has bisaturated boundary. 9.1. Smoothing out the nonideal boundary. Here is the main result of this section. Lemma 9.2. Let Γ be an inﬁnite discrete subgroup of PGL(V ) and Ω a nonempty Γ-invariant properly convex open subset of P(V ). Suppose Γ acts cor convex cocompactly on Ω. Fix a uniform neighborhood Cu of CΩ (Γ) in cor (Ω, dΩ). Then the convex core CΩ (Γ) admits a Γ-invariant, properly con- 1 vex, closed neighborhood C ⊂ Cu in Ω which has C , strictly convex nonideal boundary. Constructing a neighborhood C as in the lemma clearly involves arbitrary choices; here is one of many possible constructions, taken from [DGK3, CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 45

Lem. 6.4]. Cooper–Long–Tillmann [CLT2, Prop. 8.3] give a diﬀerent construction yielding, in the case Γ is torsion-free, a convex set C as in the lemma whose nonideal boundary has the slightly stronger property that it is locally the graph of a smooth function with positive deﬁnite Hessian.

Proof of Lemma 9.2. In this proof, we ﬁx a ﬁnite-index subgroup Γ0 of Γ which is torsion-free; such a subgroup exists by the Selberg lemma [Se, Lem. 8]. We proceed in three steps. Firstly, we construct a Γ-invariant closed neigh- cor borhood C♣ of CΩ (Γ) in Ω which is contained in Cu and whose nonideal boundary is C1 but not necessarily strictly convex. Secondly, we construct 1 a small deformation C♦ ⊂ Cu of C♣ which has C and strictly convex nonideal boundary, but which is only Γ0-invariant, not necessarily Γ-invariant.

Finally, we use an averaging procedure over translates γ ·C♦ of C♦, for γΓ0 ranging over the Γ0-cosets of Γ, to construct a Γ-invariant closed neighbor- cor 1 hood C ⊂ Cu of CΩ (Γ) in Ω which has C and strictly convex nonideal boundary.

• Construction of C♣: Consider a compact fundamental domain D for the cor cor action of Γ on CΩ (Γ). The convex hull of D in Ω is still contained in CΩ (Γ). 0 1 Let D ⊂ Cu be a closed neighborhood of this convex hull in Ω which has C 0 frontier, and let C♣ ⊂ Cu be the closure in Ω of the convex hull of Γ ·D . By orb Lemma 4.1, we have ∂iC♣ = ΛΩ (Γ) and C♣ has bisaturated boundary. 1 Let us check that C♣ has C nonideal boundary. We ﬁrst observe that any supporting hyperplane Πy of C♣ at a point y ∈ ∂nC♣ stays away from

∂iC♣ since C♣ has bisaturated boundary. On the other hand, since the action of Γ on C♣ is properly discontinuous, for any neighborhood N of ∂iC♣ in P(V ) and any inﬁnite sequence of distinct elements γj ∈ Γ, the translates 0 γj ·D are eventually all contained in N . Therefore, in a neighborhood of y, the hypersurface Fr(C♣) coincides with the convex hull of a ﬁnite union of Sm 0 1 translates i=1 γi ·D , and so it is locally C : indeed that convex hull is dual Tm 0 ∗ 0 ∗ to i=1(γi ·D ) , which has strictly convex frontier because (D ) does (a convex set has C1 frontier if and only if its dual has strictly convex frontier).

• Construction of C♦: For any y ∈ ∂nC♣, let Fy be the open stratum of Fr(C♣) at y, namely the intersection of C♣ with the unique supporting hyperplane Πy at y. Since C♣ has bisaturated boundary, Fy is a compact convex subset of ∂nC♣. We claim that Fy is disjoint from γ · Fy = Fγ·y for all γ ∈ Γ0 r {1}. 0 Indeed, if there existed y ∈ Fy ∩ Fγ·y, then by uniqueness the supporting hyperplanes would satisfy Πy = Πy0 = Πγ·y, hence Fy = Fy0 = Fγ·y = γ · Fy. m m This would imply Fy = γ · Fy for all m ∈ N, hence γ · y ∈ Fy. Using the fact that the action of Γ0 on C♣ is properly discontinuous and taking a limit, we see that Fy would contain a point of ∂iC♣, which we have seen is not true. Therefore Fy is disjoint from γ · Fy for all γ ∈ Γ0 r {1}. ∗ For any y ∈ ∂nC♣, the subset of P(V ) consisting of those projective hyperplanes near the supporting hyperplane Πy that separate Fy from ∂iC♣ is open and nonempty, hence (n−1)-dimensional where dim(V ) = n. Choose 46 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

1 n−1 i n − 1 such hyperplanes Πy,..., Πy in generic position, with Πy cutting off i i a compact region Qy ⊃ Fy from C♣. One may imagine each Πy is obtained by pushing Πy normally into C♣ and then tilting slightly in one of n − 1 Tn−1 i independent directions. The intersection i=1 Πy ⊂ P(V ) is reduced to a i singleton. By taking each hyperplane Πy very close to Πy, we may assume Sn−1 i that the union Qy := i=1 Qy is disjoint from all its γ-translates for γ ∈ 0 Γ0 r {1}. In addition, we ensure that Fy has a neighborhood Qy contained Tn−1 i in i=1 Qy. Since the action of Γ0 on ∂nC♣ is cocompact, there exist finitely many 0 0 points y1, . . . , ym ∈ ∂nC♣ such that ∂nC♣ ⊂ Γ0 · (Qy1 ∪ · · · ∪ Qym ). We now explain, for any y ∈ ∂nC♣, how to deform C♣ into a new, smaller cor 1 properly convex Γ0-invariant closed neighborhood of CΩ (Γ) in Ω with C 0 nonideal boundary, in a way that destroys all segments in Qy. Repeating for y = y1, . . . , ym, this will produce a properly convex Γ0-invariant closed cor 1 neighborhood C♦ of CΩ (Γ) in Ω with C and strictly convex nonideal boundary. Choose an affine chart containing Ω, an auxiliary Euclidean metric g + + on this chart, and a smooth strictly concave function h : R → R with d h(0) = 0 and dt t=0 h(t) = 1 (e.g. h = tanh). We may assume that i i for every 1 ≤ i ≤ n − 1 the g-orthogonal projection πy onto Πy satis- i i i i −1 i i fies π (Q ) ⊂ Π ∩ C , with (π | i ) (Π ∩ ∂C ) ⊂ Π . Define maps y y y ♣ y Qy y ♣ y i i i i i −1 0 ϕy : Qy → Qy by the property that ϕy preserves each fiber (πy) (y ) (a segment), taking the point at distance t from y0 to the point at distance h(t). i Then ϕy takes any segment σ of Fy to a strictly convex curve, unless σ is i i parallel to Πi. The image ϕy(Qy ∩ ∂nC♣) is still a convex hypersurface. Ex- i i tending ϕy by the identity on Qy r Qy and repeating with varying i, we find 1 n−1 0 that the composition ϕy := ϕy ◦· · ·◦ϕy , defined on Qy, takes Qy ∩∂nC♣ to a strictly convex hypersurface. We can extend ϕy in a Γ0-equivariant fashion to Γ0 ·Qy, and extend it further by the identity on the rest of C♣: the set cor ϕy(C♣) is still a Γ0-invariant closed neighborhood of CΩ (Γ) in Ω, contained 1 in Cu, with C nonideal boundary. Repeating with finitely many points y1, . . . , ym as above, we obtain a Γ0- cor invariant properly convex closed neighborhood C♦ ⊂ Cu of CΩ (Γ) in Ω with C1 and strictly convex boundary.

• Construction of C: Consider the ﬁnitely many Γ0-cosets γ1Γ0, . . . , γkΓ0 i i of Γ and the corresponding translates C♦ := γi ·C♦; we denote by Ω the i 0 interior of C♦. Let C be a Γ-invariant properly convex closed neighborhood cor 1 of CΩ (Γ) in Ω which has C (but not necessarily strictly convex) nonideal i 0 boundary and is contained in all C♦, 1 ≤ i ≤ k. (Such a neighborhood C can be constructed for instance by the same method as C♣ above.) Since i 0 C♦ has strictly convex nonideal boundary, uniform neighborhoods of C in i (Ω , dΩi ) have strictly convex nonideal boundary [Bu, (18.12)]. Therefore, by cocompactness, if h : [0, 1] → [0, 1] is a convex function with suﬃciently CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 47

α fast growth (e.g. h(t) = t for large enough α > 0), then the Γ0-invariant 0 −1 function Hi := h ◦ dΩi (·, C ) is convex on the convex region Hi ([0, 1]), and in fact smooth and strictly convex near every point outside C0. The function Pk −1 H := i=1 Hi is Γ-invariant and its sublevel set C := H ([0, 1]) is a Γ- cor 1 invariant closed neighborhood of CΩ (Γ) in Ω which has C , strictly convex nonideal boundary. Moreover, C ⊂ C♦ ⊂ C♣ ⊂ Cu by construction. 9.2. Proof of the remaining implications of Theorems 1.15 and 1.20.

Proof of (1) ⇒ (4) in Theorem 1.20. Suppose Γ is convex cocompact in P(V ), i.e. it preserves a properly convex open set Ω ⊂ P(V ) and acts cocompactly cor on the convex core CΩ (Γ). Let Cu be the closed uniform 1-neighborhood cor of CΩ (Γ) in (Ω, dΩ). The set Cu is properly convex [Bu, (18.12)]. The action of Γ on Cu is properly discontinuous, and cocompact since Cu is the union of the Γ-translates of the closed uniform 1-neighborhood of a compact cor cor fundamental domain of CΩ (Γ) in (Ω, dΩ). By Lemma 9.2, the set CΩ (Γ) admits a Γ-invariant, properly convex, closed neighborhood Csmooth ⊂ Cu in Ω 1 which has C , strictly convex nonideal boundary. The action of Γ on Csmooth orb is still properly discontinuous and cocompact, and ∂iCsmooth = ΛΩ (Γ) by Lemma 4.1.(1). This proves the implication (1) ⇒ (4) in Theorem 1.20. Proof of (iii) ⇒ (i) in Theorem 1.15. Suppose that Γ acts convex cocompactly on a properly convex open set Ω and that the full orbital limit set orb ΛΩ (Γ) contains no nontrivial segment. As in the proof of (1) ⇒ (4) in The- orem 1.20 just above, Γ acts properly discontinuously and cocompactly on some nonempty closed properly convex subset Csmooth of Ω which has strictly 1 convex and C nonideal boundary and whose interior Ωsmooth := Int(Csmooth) cor contains CΩ (Γ). The set Csmooth has bisaturated boundary (Lemma 9.1), hence the action of Γ on Ωsmooth is convex cocompact by Corollary 4.4. orb By Lemma 4.1.(1), the ideal boundary ∂iCsmooth is equal to ΛΩ (Γ). Since this set contains no nontrivial segment by assumption, we deduce from Lemma 4.1.(1) that any z ∈ ∂iCsmooth is an extreme point of ∂Ω. Thus the full boundary of Ωsmooth is strictly convex. ∗ ∗ Consider the dual convex set Csmooth ⊂ P(V ) (Deﬁnition 5.1); it is prop- ∗ erly convex. By Lemma 5.3, the set Csmooth has bisaturated boundary and does not contain any PET (since Csmooth itself does not contain any PET). ∗ By Proposition 5.4, the action of Γ on Csmooth is properly discontinuous and ∗ cocompact. It follows from Lemma 6.2 that ∂iCsmooth contains no nontrivial ∗ segment, hence each point of ∂iCsmooth is an extreme point. Hence there is exactly one hyperplane supporting Csmooth at any given point of ∂iCsmooth. This is also true at any given point of ∂nCsmooth by assumption. Thus the 1 boundary of Ωsmooth is C . 10. Properties of convex cocompact groups In this section we prove Theorem 1.17. Property (A) has already been established in Section 5.3; we now establish the other properties. 48 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

10.1. (B): Quasi-isometric embedding. We now establish the following very general result, using the notation µ1 − µn from (2.1). The fact that a statement of this ﬂavor should exist was suggested to us by Yves Benoist.

Proposition 10.1. Let Ω be a properly convex subset of P(V ). For any z ∈ Ω, there exists κ > 0 such that for any g ∈ Aut(Ω),

(µ1 − µn)(g) ≥ 2 dΩ(z, g · z) − κ. Here is an easy consequence of Proposition 10.1.

Corollary 10.2. Let Γ be a discrete subgroup of G := PGL(V ). If Γ is naively convex cocompact in P(V ) (Definition 1.9), then it is finitely generated and the natural inclusion Γ ,→ G is a quasi-isometric embedding. In particular, for any subgroup Γ0 of Γ which is naively convex cocompact 0 in P(V ), the natural inclusion Γ ,→ Γ is a quasi-isometric embedding. Here the finitely generated group Γ is endowed with the word metric with respect to some fixed finite generating subset. The group G is endowed with any G-invariant Riemannian metric.

Proof of Corollary 10.2. Let K ' PO(n) be a maximal compact subgroup of G = PGL(V ) as in Section 2.2. Let p := eK ∈ G/K. There is a G-invariant metric d on G/K such that d(p, g · p) = (µ1 − µn)(g) for all g ∈ G. Let Ω be a Γ-invariant properly convex open subset of P(V ) and C a nonempty Γ-invariant closed convex subset of Ω on which Γ acts cocompactly. By the Švarc–Milnor lemma, Γ is ﬁnitely generated and any orbital map Γ → (C, dΩ) is a quasi-isometry. Using Proposition 10.1, we then obtain that any orbital map Γ → (G/K, d) is a quasi-isometric embedding. Since K is compact, the natural inclusion Γ ,→ G is a quasi-isometric embedding.

In order to prove Proposition 10.1, we ﬁrst establish the following estimate, where we consider the metric 0 0 dP([x], [x ]) := | sin ](x, x )| on P(V ) as in Section 8.2.

Lemma 10.3. Let Ω be a properly convex open subset of P(V ). For any z ∈ Ω, there exists Rz ≥ 1 such that for any w ∈ Ω and b ∈ ∂Ω with z, w, b aligned in this order, −1 Rz ≤ dΩ(z, w) · dP(w, b) ≤ Rz. n−1 Proof of Lemma 10.3. Fix z ∈ Ω. Consider an aﬃne chart R of P(V ), endowed with a Euclidean norm k · k, in which Ω appears nested between two Euclidean balls of radii r < R centered at z. There exists B > 0 such that for any y1, y2 ∈ Ω, −1 B dP(y1, y2) ≤ ky1 − y2k ≤ B dP(y1, y2). CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 49

For any w ∈ Ω and a, b ∈ ∂Ω with a, z, w, b aligned in this order, we then have r2 ka˜ − w˜k kz˜ − ˜bk 2R2 ≤ ≤ , ˜ −1 R · B dP(w, b) ka˜ − z˜k kw˜ − bk r · B dP(w, b) and the middle term is equal to dΩ(z, w) by deﬁnition of dΩ.

˜ Proof of Proposition 10.1. Let k · kV be a K-invariant Euclidean norm on V for which (2.2) holds. Fix z ∈ Ω. We ﬁrst make the observation that by compactness of ∂Ω, there exists 0 < ε < 1/2 with the following property: 0 for any lift z˜ ∈ V r {0} of z and any segment [x, x ] ⊂ V r {0} containing z˜ and projecting to a segment of Ω ⊂ P(V ) with distinct endpoints in ∂Ω, we 0 0 0 have dP([x], [x ])| ≥ ε, and if kxkV = kx kV we can write z˜ = tx + (1 − t)x where ε ≤ t ≤ 1 − ε. Fix g ∈ G and lift it to an element of SL±(V ), still denoted by g. Consider α, β ∈ ∂Ω such that g · α, z, g · z, g · β are aligned in this order. Lift α, β, z ˜ ˜ ˜ respectively to α,˜ β, z˜ ∈ V r {0} such that kα˜kV = kβkV and z˜ ∈ [˜α, β]. By the above observation, we have dP(g · α, g · β]) ≥ ε and we may write z˜ = tα˜ + (1 − t)β˜ with ε ≤ t ≤ 1 − ε. We have ˜ ˜ kg · z˜kV kg · βkV dP(g · z, g · β) = area (g · z,˜ g · β) = t area (g · α,˜ g · β˜) ˜ = t kg · α˜kV kg · βkV dP(g · α, g · β), by deﬁnition of dP, where area (v, w) denotes the area of the parallelogram of V spanned by the vectors v, w. Therefore

kg · z˜kV 2 −1 ≥ ε dP(g · z, g · β) . kg · α˜kV

Let Rz > 0 be given by Lemma 10.3. Taking (w, b) = (g · z, g · β) in Lemma 10.3, we have

kg · z˜kV 2 −1 (10.1) ≥ ε Rz dΩ(z, g · z). kg · α˜kV −1 ± −1 Let kgkV (resp. kg kV ) be the operator norm of g ∈ SL (V ) (resp. g ∈ SL±(V )). Then

kg · z˜kV ≤ kgkV kzkV ≤ kgkV kαkV and kαkV kg · α˜kV ≥ −1 . kg kV Therefore, taking the logarithm in (10.1) and using (2.2), we obtain −1 2 −1 (µ1 − µn)(g) = log kgkV kg kV ≥ 2 dΩ(z, g · z) − | log(ε Rz )|.

10.2. (C): No unipotent elements. Property (C) of Theorem 1.17 is contained in the following more general statement. 50 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Proposition 10.4. Let Γ be a discrete subgroup of PGL(V ) which is naively convex cocompact in P(V ) (Deﬁnition 1.9). Then Γ does not contain any unipotent element.

Proof. Let Ω be a Γ-invariant properly convex open subset of P(V ) and C a nonempty Γ-invariant closed convex subset of Ω on which Γ acts cocompactly. By [CLT1, Prop. 2.13], we only need to check that any element γ ∈ Γ of inﬁnite order achieves its translation length

R := inf dΩ(y, γ · y) ≥ 0. y∈C

Consider (ym) ∈ CN such that dΩ(ym, γ · ym) → R. For any m, there exists γm ∈ Γ such that γm ·ym belongs to some ﬁxed compact fundamental domain for the action of Γ on C. Up to passing to a subsequence, γm ·ym converges to some y ∈ C. As m → +∞, we have dΩ(ym, γ · ym) = R + o(1), which means −1 that γmγγm ∈ Γ sends γm · ym ∈ C at distance ≤ R + o(1) from itself. Since the action of Γ on C is properly discontinuous, we deduce that the discrete −1 sequence (γmγγm )m∈N is bounded; up to passing to a subsequence, we may −1 therefore assume that it is constant, equal to γ∞γγ∞ for some γ∞ ∈ Γ. We −1 then have dΩ(z, γ · z) = R where z := γ∞ · y.

10.3. (D): Stability. Property (D) of Theorem 1.17 follows from the equivalence (2) ⇔ (4) of Theorem 1.20 and from the work of Cooper–Long– Tillmann (namely [CLT2, Th. 0.1] with empty collection V of generalized cusps). Indeed, the equivalence (2) ⇔ (4) of Theorem 1.20 shows that Γ is convex cocompact in P(V ) if and only if it acts properly discontinuously and cocompactly on a properly convex subset C of P(V ) with strictly convex nonideal boundary, and this condition is stable under small perturbations by [CLT2].

10.4. (E): Semisimpliﬁcation. Let Γ be a discrete subgroup of PGL(V ). The Zariski closure H of Γ in PGL(V ) admits a Levi decomposition H = L n Ru(H) where L is reductive (called a Levi factor) and Ru(H) is the unipotent radical of H. The projection of Γ to L is discrete (see [R, Th. 8.24]) and does not depend, up to conjugation in PGL(V ), on the choice of the Levi factor L. We shall use the following terminology.

Definition 10.5. The semisimplification of Γ is the projection of Γ to a Levi factor of the Zariski closure of Γ. It is a discrete subgroup of PGL(V ), well defined up to conjugation.

The semisimplification of Γ, like the Levi factor containing it, acts projectively on V in a semisimple way: namely, any invariant linear subspace of V admits an invariant complementary subspace. Suppose that the semisimplification of Γ is convex cocompact in P(V ). This semisimplification is a limit of PGL(V )-conjugates of Γ. Since convex CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 51 cocompactness is open (property (D) above) and invariant under conjugation, we deduce that Γ is convex cocompact in P(V ). This proves property (E) of Theorem 1.17.

0 n0 10.5. (F): Inclusion into a larger space. Let V = R and let j : V,→ V ⊕ V 0 j∗ : V ∗ ,→ (V ⊕ V 0)∗ ' V ∗ ⊕ (V 0)∗ be the natural inclusions. We use the same letters for the induced inclusions of projective spaces. Let i : SL±(V ) ,→ SL±(V ⊕V 0) be the natural inclusion, whose image acts trivially on the second factor. Suppose Γ ⊂ PGL(V ) acts convex cocompactly on some nonempty properly convex open subset Ω of P(V ). By Proposition 5.6, we may choose Ω so ∗ ∗ that the dual action of Γ on the dual convex set Ω ⊂ P(V ) is also convex cocompact. Let Γˆ be the lift of Γ to SL±(V ) that preserves a properly convex cone of V lifting Ω (see Remark 3.1).

0 Lemma 10.6. (1) Let K be a compact subset of P(V ⊕ V ) that does not meet any projective hyperplane z∗ ∈ j∗(Ω∗). Then any accumulation 0 ˆ orb point in P(V ⊕ V ) of the i(Γ)-orbit of K is contained in j(ΛΩ (Γ)). ∗ 0 ∗ (2) Let K be a compact subset of P((V ⊕V ) ) whose elements correspond 0 to projective hyperplanes disjoint from j(Ω) in P(V ⊕ V ). Then any 0 ∗ ∗ accumulation point in P((V ⊕V ) ) of the i(Γ)ˆ -orbit of K is contained ∗ orb in j (ΛΩ∗ (Γ)).

Proof. The two statements are dual to each other, so we only need to prove (1). 0 0 Let π : P(V ⊕V )rP(V ) → P(V ) be the map induced by the projection onto 0 0 the first factor of V ⊕ V , so that π ◦ j is the identity of P(V ). Note that V 0 is contained in every projective hyperplane of P(V ⊕V ) corresponding to an ∗ ∗ element of j (Ω ), hence π is defined on K. Consider a sequence (zm)m∈N of ˆ points of K and a sequence (γm)m∈N of pairwise disjoint elements of Γ such 0 that (i(γm)·zm)m∈N converges in P(V ⊕V ). By construction, for any m, the ∗ ∗ ∗ point zm does not belong to any hyperplane z ∈ j (Ω ), hence π(zm) does ∗ not belong to any hyperplane in Ω , i.e. π(zm) ∈ Ω. Since Γˆ acts properly discontinuously on Ω, the point y∞ := limm γm·π(zm) is contained in ∂Ω. Up to passing to a subsequence, we may assume that (π(zm))m∈N ⊂ π(K) converges to some y ∈ Ω. Then dΩ(γm · π(zm), γm · y) = dΩ(π(zm), y) → 0, and orb so γm · y → y∞ by Corollary 3.5. In particular, y∞ ∈ ΛΩ (Γ). Now lift zm to 0 0 0 0 a vector xm+xm ∈ V ⊕V , with (xm)m∈N ⊂ V ⊕{0} and (xm)m∈N ⊂ {0}⊕V bounded. The image of xm in P(V ) is π(zm). By Lemma 3.2, the sequence 0 0 (i(γm) · xm)m∈N tends to infinity in V . On the other hand, i(γm) · xm = xm remains bounded in V 0. Therefore, orb lim i(γm) · zm = lim i(γm) · j ◦ π(zm) = lim j(γm · π(zm)) ∈ j(Λ (Γ)). m m m Ω 52 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

The set j(Ω) is contained in the Γ-invariant open subset 0 [ ∗ Omax := P(V ⊕ V ) r z z∗∈j∗(Ω∗) 0 of P(V ⊕ V ), which is convex but not properly convex. This set Omax is 0 the union of all projective lines of P(V ⊕ V ) intersecting both j(Ω) and 0 P({0} × V ). Lemma 10.7. The set j(Ω) is contained in a Γ-invariant properly convex open set O ⊂ Omax. Proof. We argue similarly to the proof of Lemma 8.5. Choose projective 0 hyperplanes H0,...,HN of P(V ⊕V ) bounding an open simplex ∆ containing j(Ω). Let U := ∆ ∩ Omax. Deﬁne \ O := i(γ) ·U. γ∈Γˆ

Then j(Ω) ⊂ O ⊂ Omax and O is properly convex. We claim that O is open. Indeed, suppose by contradiction that there exists a point z ∈ ∂nO. Then there is a sequence (γm) in Γˆ and k ∈ {0,...,N} such that γm · Hk converges to a hyperplane H∞ containing z. This is impossible since, ∗ orb by Lemma 10.6.(2), the hyperplane H∞ ∈ j (ΛΩ (Γ)) supports the open set Omax which contains z. orb ˆ orb Finally, observe that ΛO (i(Γ)) = j(ΛΩ (Γ)) by Lemma 10.6.(1). Hence ˆ cor ˆ cor the action of i(Γ) on CO (i(Γ)) = j(CΩ (Γ)) is cocompact and we conclude 0 that i(Γ)ˆ is convex cocompact in P(V ⊕ V ). 10.6. A consequence of properties (A), (D), and (F) of Theorem 1.17. The following result uses reasoning similar to Section 10.4. Proposition 10.8. Let Γ be a discrete subgroup of SL±(V ) acting trivially on some linear subspace V0 of V . Then the induced action of Γ on P(V/V0) is convex cocompact if and only if the action of Γ on P(V ) is convex cocompact.

Proof. Throughout the proof we fix a complement V1 to V0 in V so that V = V0 ⊕ V1. We first use properties (D) and (F) to prove the forward implication. We can write all elements γ ∈ Γ as upper triangular block matrices with respect to the decomposition V = V0 ⊕ V1: IB [γ] = . V0⊕V1 0 A Let Γdiag be the discrete subgroup of PGL(V ) obtained by considering only diag the two diagonal blocks. The action of Γ on V1 identifies with the action diag of Γ on V/V0. Therefore the action of Γ on P(V1) is convex cocompact. diag By property (F) above, the action of Γ on P(V ) is convex cocompact. diag −1 We now note that Γ is the limit of the conjugates gmΓgm of Γ, where gm is a diagonal matrix acting on V0 by 1/m and on V1 by m. Since convex CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 53 cocompactness is open (property (D)), we deduce that the action of Γ on P(V ) is convex cocompact. We now check the converse implication. Suppose Γ is convex cocompact in P(V ). By property (A) of Theorem 1.17, the group Γ is also convex ∗ cocompact in the dual P(V ), and it will be easiest to work there. There is ∗ ∗ ∗ ∗ ∗ a dual splitting V = V0 ⊕ V1 , where V0 and V1 are the linear functionals ∗ which vanish on V1 and V0 respectively. Note that V1 is independent of the ∗ choice of the complement V1 to V0 in V , hence V1 is invariant under Γ. The matrix of γ ∈ Γ for the action on V ∗ with respect to the concatenation of a ∗ ∗ basis for V0 and a basis for V1 has the form I 0 [γ]V ∗⊕V ∗ = t t . 0 1 Bγ Aγ By Lemma 3.2, the accumulation points of the orbit of any compact set of ∗ ∗ ∗ ∗ points in P(V ) r P(V0 ) lie in P(V1 ). Let C be a properly convex subset ∗ of P(V ) with bisaturated boundary on which the action of Γ is properly ∗ discontinuous and cocompact. Then the ideal boundary ∂iC is contained ∗ ∗ ∗ ∗ in P(V1 ). The intersection C ∩ P(V1 ) is a properly convex subset of P(V1 ) with bisaturated boundary and the action of Γ is properly discontinuous and ∗ cocompact. Hence the restriction of Γ is convex cocompact in P(V1 ), and so the restriction of Γ (for the dual action, with the matrix of γ being Aγ) is convex cocompact in P(V1). We now conclude in the same way as for the forward implication using property (F) of Theorem 1.17. 10.7. A consequence of properties (E) and (F) of Theorem 1.17. Proposition 10.9. Let V 00 = V ⊕ V 0 be a direct sum of finite-dimensional real vector spaces, and Γ a discrete subgroup of SL±(V ) ⊂ SL±(V 00). Any map u :Γ → Hom(V 0,V ) satisfying the cocycle relation

u(γ1γ2) = u(γ1) + γ1 · u(γ2) u ± 00 00 for all γ1, γ2 ∈ Γ defines a discrete subgroup Γ of SL (V ), hence of PGL(V ). u 00 If Γ is convex cocompact in P(V ), then Γ is convex cocompact in P(V ). Recall that u :Γ → Hom(V 0,V ) is said to be a coboundary if there exists ϕ ∈ Hom(V 0,V ) such that u(γ) = ϕ − γ · ϕ for all γ ∈ Γ. This is equivalent to Γu being conjugate to a subgroup of SL±(V ). Remark 10.10. If the cocycle u is not a coboundary, then Γu is a discrete 00 00 subgroup of PGL(V ) which is convex cocompact in P(V ) but not com- pletely reducible. Such a cocycle u always exists e.g. if Γ is a free group. See also Example 10.11. 3 Example 10.11. We briefly examine the special case where V = R , where 0 1 3 V = R , and where Γ is a discrete subgroup of PO(2, 1) ⊂ PGL(R ) isomorphic to the fundamental group of a closed orientable surface Sg of genus g ≥ 2. In this case, Γ represents a point of the Teichmüller space T (Sg) and the space of cohomology classes of cocycles u :Γ → Hom(V 0,V ) ' V 54 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL identifies with the tangent space to T (Sg) at Γ (see e.g. [DGK1, § 2.3]) and has dimension 6g − 6. (1) By Proposition 10.9, the group Γu preserves and acts convex cocom- 3 1 pactly on a nonempty properly convex open subset Ω of P(R ⊕ R ). The u Γ -invariant convex open set Ωmax ⊃ Ω given by Proposition 4.5.(3) turns out to be properly convex as soon as u is not a coboundary. It can be described u 3 3 1 as follows. The group Γ preserves an affine chart U ' R of P(R ⊕R ) and a flat Lorentzian metric on U. The action on U is not properly discontinuous. However, Mess [Me] described two maximal globally hyperbolic domains of discontinuity, called domains of dependence, one oriented to the future and one oriented to the past. The domain Ωmax is the union of the two domains 2 3 of dependence plus a copy of the hyperbolic plane H in P(R ⊕ 0). We have Λorb (Γu) ⊂ ∂ 2 ⊂ ( 3 ⊕ 0) and Ccor (Γu) ⊂ 2. Ωmax H P R Ωmax H (2) By Theorem 1.17.(A), the group Γu also acts convex cocompactly on a nonempty properly convex open subset Ω∗ of the dual projective space 3 ∗ u ∗ ∗ P((R ⊕ R) ). In this case the Γ -invariant convex open set (Ω )max ⊃ Ω , given by Proposition 4.5.(3) applied to Ω∗, is not properly convex: it is 3 2 ∗ 3 ∗ the suspension HP of the dual copy (H ) ⊂ P((R ) ) of the hyperbolic 1 ∗ 3 3 ∗ plane with the point P((R ) ). The convex set HP ⊂ P((R ⊕ R) ) is the projective model for half-pipe geometry, a transitional geometry lying between hyperbolic geometry and anti-de Sitter geometry (see [Dan]). The orb ∗ 3 full orbital limit set ΛΩ∗ (Γ) = Ω ∩ ∂HP is not contained in a hyperplane cor u if u is not a coboundary. Indeed, CΩ∗ (Γ ) may be thought of as a rescaled limit of the collapsing convex cores for quasi-Fuchsian subgroups of PO(3, 1) (or of PO(2, 2)) which converge to the Fuchsian group Γ. This situation was described in some detail by Kerckhoff in a 2010 lecture at the workshop on Geometry, topology, and dynamics of character varieties at the National University of Singapore, and other lectures in 2011 and 2012 about the work [DK] (still in preparation).

Proof of Proposition 10.9. Suppose Γ is convex cocompact in P(V ). Then its 00 00 image i(Γ) in PGL(V ) is convex cocompact in P(V ) by Theorem 1.17.(F). But i(Γ) is the semisimpliﬁcation of Γu. Therefore Γu is convex cocompact 00 in P(V ) by Theorem 1.17.(E).

p,q−1 11. Convex cocompactness in H ∗ p+q Fix p, q ∈ N . Recall from Section 1.8 that the projective space P(R ) is the disjoint union of p,q−1 p,q H = [x] ∈ P(R ) | hx, xip,q < 0 , p−1,q p,q of S = {[x] ∈ P(R ) | hx, xip,q > 0}, and of p,q−1 p−1,q p,q ∂H = ∂S = [x] ∈ P(R ) | hx, xip,q = 0 . For instance, Figure 5 shows 4 3,0 3,0 2,1 2,1 P(R ) = H t ∂H = ∂S t S CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 55 and 4 2,1 2,1 1,2 1,2 P(R ) = H t ∂H = ∂S t S . p,q−1 As explained in [DGK3, § 2.1], the space H has a natural PO(p, q)-

` `1 `0

S2,1 S1,2 `2 H3,0 H2,1

3,0 Figure 5. Left: H with a geodesic line ` (necessarily 2,1 2,1 spacelike), and S . Right: H with three geodesic lines 1,2 `2 (spacelike), `1 (lightlike), and `0 (timelike), and S . invariant pseudo-Riemannian structure of constant negative curvature, for p,q−1 which the geodesic lines are the intersections of H with projective lines p,q in P(R ). Such a line is called spacelike (resp. lightlike, resp. timelike) if it p,q−1 meets ∂PH in two (resp. one, resp. zero) points: see Figure 5. p+q Remark 11.1. An element g ∈ PO(p, q) is proximal in P(R ) (Defini- + p,q−1 tion 2.1) if and only if it admits a unique attracting fixed point ξg in ∂H , p,q−1 in which case we shall say that g is proximal in ∂H . In particular, for a p+q discrete subgroup Γ of PO(p, q) ⊂ PGL(R ), the proximal limit set ΛΓ of p,q p,q−1 Γ in P(R ) (Definition 2.2) is contained in ∂H , and called the proximal p,q−1 limit set of Γ in ∂H . In this section we prove Theorems 1.25 and 1.29. For Theorem 1.25, the equivalences (7) ⇔ (8) ⇔ (9) ⇔ (10) follow from the equivalences (3) ⇔ (4) ⇔ (5) ⇔ (6) by replacing the symmetric bilinear form h·, ·ip,q by −h·, ·ip,q ' h·, ·iq,p: see [DGK3, Rem. 4.1]. The implication (3) ⇒ (6) is contained in the forward implication of Theorem 1.4, which has been established in Section 7. Here we shall prove the implications (6) ⇒ (5) ⇒ (4) ⇒ (3) and (1) ⇒ (2). We note that (2) ⇒ (1) is contained in (4) ⇒ (3) and (8) ⇒ (7). 11.1. Proof of the implication (6) ⇒ (5) in Theorem 1.25. Suppose p,q that Γ is word hyperbolic, that the natural inclusion Γ ,→ PO(p, q) is P1 - p,q−1 Anosov, and that the proximal limit set ΛΓ ⊂ ∂H is negative. p,q By definition of negativity, the set ΛΓ lifts to a cone ΛeΓ of R r {0} on which all inner products h·, ·ip,q of noncollinear points are negative. The set p,q 0 0 Ωmax := P x ∈ R | hx, x ip,q < 0 ∀x ∈ ΛeΓ 56 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL is a connected component of ( p,q) S z⊥ which is open and convex. P R r z∈ΛΓ It is nonempty because it contains x1 + x2 for any noncollinear x1, x2 ∈ ΛeΓ. p,q Note that if ΛbΓ is another cone of R r {0} lifting ΛΓ on which all inner products h·, ·ip,q of noncollinear points are negative, then either ΛbΓ = ΛeΓ or ΛbΓ = −ΛeΓ (using negativity). Thus Ωmax is well defined independently of the lift ΛeΓ, and so Ωmax is Γ-invariant because ΛΓ is. By Proposition 8.1, the group Γ acts convex cocompactly on some non- orb empty, properly convex open subset Ω ⊂ Ωmax, and ΛΩ (Γ) = ΛΓ. In cor particular, the convex hull CΩ (Γ) of ΛΓ in Ω has compact quotient by Γ. cor p,q−1 As in [DGK3, Lem. 3.6.(1)], this convex hull CΩ (Γ) is contained in H : cor Pk indeed, any point of CΩ (Γ) lifts to a vector of the form x = i=1 tixi with k ≥ 2, where x1, . . . , xk ∈ ΛeΓ are distinct and t1, . . . , tk > 0; for all i 6= j we have hxi, xiip,q = 0 and hxi, xjip,q < 0 by negativity of ΛΓ, hence hx, xip,q < 0. cor Since CΩ (Γ) has compact quotient by Γ, it is easy to check that for small cor enough r > 0 the open uniform r-neighborhood U of CΩ (Γ) in (Ω, dΩ) is p,q−1 contained in H and properly convex (see [DGK3, Lem. 6.3]). The full orb orb orbital limit set ΛU (Γ) is a nonempty closed Γ-invariant subset of ΛΩ (Γ), cor hence its convex hull CU (Γ) in U is a nonempty closed Γ-invariant subset cor cor cor of CΩ (Γ). (In fact CU (Γ) = CΩ (Γ) by Lemma 4.1.(1).) The nonempty cor set CU (Γ) has compact quotient by Γ. Thus Γ acts convex cocompactly on p,q−1 U ⊂ H , and ΛΓ is transverse since it is negative.

11.2. Proof of the implication (5) ⇒ (4) in Theorem 1.25. Suppose that Γ acts convex cocompactly on some nonempty properly convex open p,q−1 subset Ω of H and that the proximal limit set ΛΓ is transverse. The set orb ΛΓ is contained in ΛΩ (Γ). By Lemma 4.1.(1)–(2), the convex hull of ΛΓ in Ω cor orb cor is the convex hull CΩ (Γ) of ΛΩ (Γ) in Ω, and the ideal boundary ∂iCΩ (Γ) orb p,q−1 is equal to ΛΩ (Γ). By Corollary 3.3, this set is contained in ∂H . It orb orb follows that ΛΓ = ΛΩ (Γ): indeed, otherwise ΛΩ (Γ) would contain a nontrivial segment between two points of ΛΓ, this segment would be contained p,q−1 in ∂H , and this would contradict the assumption that ΛΓ is transverse. cor p,q−1 Thus CΩ (Γ) is a closed properly convex subset of H on which Γ acts properly discontinuously and cocompactly, and whose ideal boundary does not contain any nontrivial projective line segment. cor It could be the case that CΩ (Γ) has empty interior. However, for any r > 0 cor the closed uniform r-neighborhood Cr of CΩ (Γ) in (Ω, dΩ) has nonempty interior, and is still properly convex with compact quotient by Γ. By Lem- orb ma 4.1.(1), we have ∂iCr = ΛΩ (Γ), hence ∂iCr does not contain any nontriv- p,q−1 ial projective line segment. This shows that Γ is H -convex cocompact.

11.3. Proof of the implication (1) ⇒ (2) in Theorem 1.25. The proof relies on the following proposition, which is stated in [DGK3, Prop. 3.7] for CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 57 irreducible Γ. The proof for general Γ is literally the same; we recall it for the reader’s convenience. ∗ Proposition 11.2. For p, q ∈ N , let Γ be a discrete subgroup of PO(p, q) p,q preserving a nonempty properly convex open subset Ω of P(R ). Let ΛΓ ⊂ p,q−1 ∂H be the proximal limit set of Γ (Deﬁnition 2.2 and Remark 11.1). If ΛΓ contains at least two points and is transverse, and if the action of Γ on ΛΓ is minimal (i.e. every orbit is dense), then ΛΓ is negative or positive.

Recall from Remark 2.3 that if Γ is irreducible with ΛΓ 6= ∅, then the action of Γ on ΛΓ is always minimal.

Proof. Suppose that ΛΓ contains at least two points and is transverse, and that the action of Γ on ΛΓ is minimal. By Proposition 4.5.(2), the sets Ω and ΛΓ lift to cones Ωe and ΛeΓ of V r {0} with Ωe properly convex containing ∗ ∗ ∗ ∗ ∗ ΛeΓ in its boundary, and Ω and ΛΓ lift to cones Ωe and ΛeΓ of V r {0} ∗ ∗ with Ωe properly convex containing ΛeΓ in its boundary, such that `(x) ≥ 0 ∗ for all x ∈ ΛeΓ and ` ∈ ΛeΓ. By Remark 3.1, the group Γ lifts to a discrete ˆ ∗ ∗ subgroup Γ of O(p, q) preserving Ωe (hence also ΛeΓ, Ωe , and ΛeΓ). Note that p,q p,q ∗ the map ψ : x 7→ hx, ·ip,q from R to (R ) induces a homeomorphism ∗ ∗ ∗ + − ΛΓ ' ΛΓ. For any x ∈ ΛeΓ we have ψ(x) ∈ ΛeΓ ∪ −ΛeΓ. Let F (resp. F ) be ∗ the subcone of ΛeΓ consisting of those vectors x such that ψ(x) ∈ ΛeΓ (resp. ∗ + 0 ψ(x) ∈ −ΛeΓ). By construction, we have x ∈ F if and only if hx, x ip,q ≥ 0 0 + ˆ for all x ∈ ΛeΓ; in particular, F is closed in ΛeΓ and Γ-invariant. Similarly, − + − F is closed and Γˆ-invariant. The sets F and F are disjoint since ΛΓ contains at least two points and is transverse. Thus F + and F − are disjoint, ˆ p,q Γ-invariant, closed subcones of ΛeΓ, whose projections to P(R ) are disjoint, Γ-invariant, closed subsets of ΛΓ. Since the action of Γ on ΛΓ is minimal, p,q ΛΓ is the smallest nonempty Γ-invariant closed subset of P(R ), and so + − + {F ,F } = {ΛeΓ, ∅}. If ΛeΓ = F then ΛΓ is nonnegative, hence positive by − transversality. Similarly, if ΛeΓ = F then ΛΓ is negative. We can now prove the implication (1) ⇒ (2) in Theorem 1.25. p+q Suppose Γ is strongly convex cocompact in P(R ). By the implication (i) ⇒ (vi) of Theorem 1.15, the group Γ is word hyperbolic and the natural p,q inclusion Γ ,→ PO(p, q) is P1 -Anosov. If #ΛΓ > 2, then the action of Γ on ∂∞Γ, hence on ΛΓ, is minimal, and so Proposition 11.2 implies that the set ΛΓ is negative or positive. This last conclusion also holds, vacuously, if #ΛΓ ≤ 2. Thus the implications (6) ⇒ (4) and (10) ⇒ (8) of Theorem 1.25 p,q−1 (proved in Sections 11.1 and 11.2 just above) show that Γ is H -convex co- q,p−1 compact or H -convex cocompact. This completes the proof of (1) ⇒ (2) in Theorem 1.25. 11.4. Proof of the implication (4) ⇒ (3) in Theorem 1.25. Suppose p,q−1 Γ ⊂ PO(p, q) is H -convex cocompact, i.e. it acts properly discontinu- p,q−1 ously and cocompactly on a closed convex subset C of H such that C has 58 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL nonempty interior and ∂iC does not contain any nontrivial projective line segment. We shall first show that Γ satisfies condition (v) of Theorem 1.15: p,q namely, Γ preserves a nonempty properly convex open subset Ω of P(R ) and acts cocompactly on a closed convex subset C0 of Ω with nonempty 0 interior, such that ∂iC does not contain any nontrivial segment. One naive idea would be to take Ω = Int(C) and C0 to be the convex hull C0 of ∂iC in C (or some small thickening), but C0 might not be contained in Int(C) (e.g. if C = C0). So we must find a larger open set Ω. Lemma 11.3 below implies that the Γ-invariant convex open subset Ωmax ⊃ Int(C) of Proposition 4.5 contains C. If Ωmax is properly convex (e.g. if Γ is irre- 0 ducible), then we may take Ω = Ωmax and C = C. However, Ωmax might not be properly convex; we shall show (Lemma 11.4) that the Γ-invariant properly convex open set Ω = Int(C)∗ (realized in the same projective space via the quadratic form) contains C0 (though possibly not C) and we shall 0 take C to be the intersection of C with a neighborhood of C0 in Ω. The following key observation is similar to [DGK3, Lem. 4.2].

∗ Lemma 11.3. For p, q ∈ N , let Γ be a discrete subgroup of PO(p, q) acting properly discontinuously and cocompactly on a nonempty properly convex p,q−1 ⊥ closed subset C of H . Then C does not meet any hyperplane z with z ∈ ∂iC. In other words, any point of C sees any point of ∂iC in a spacelike direction.

⊥ Proof. Suppose by contradiction that C meets z for some z ∈ ∂iC. Then ⊥ z contains a ray [y, z) ⊂ ∂nC. Let (am)m∈N be a sequence of points of [y, z) converging to z (see Figure 6). Since Γ acts cocompactly on C, for any m there exists γm ∈ Γ such that γm · am belongs to a ﬁxed compact subset of ∂nC. Up to taking a subsequence, the sequences (γm · am)m and (γm · y)m p,q and (γm · z)m converge respectively to some points a∞, y∞, z∞ in P(R ). We have a∞ ∈ ∂nC and y∞ ∈ ∂iC (because the action of Γ on C is properly p,q−1 discontinuous) and z∞ ∈ ∂iC (because ∂iC = Fr(C) ∩ ∂H is closed in p,q ⊥ P(R )). The segment [y∞, z∞] is contained in z∞, hence its intersection p,q−1 p,q−1 with H is contained in a lightlike geodesic and can meet ∂H only p,q at z∞. Therefore y∞ = z∞ and the closure of C in P(R ) contains a full projective line, contradicting the proper convexity of C.

Lemma 11.4. In the setting of Lemma 11.3, suppose C has nonempty interior. Then the convex hull C0 of ∂iC in C is contained in the Γ-invariant properly convex open set p,q ⊥ Ω := y ∈ P(R ) | y ∩ C = ∅ . p,q p,q ∗ Note that Ω is the dual of Int(C) realized in R (rather than (R ) ) via the symmetric bilinear form h·, ·ip,q.

p,q−1 Proof. The properly convex set C ⊂ H lifts to a properly convex cone p,q Ce of R r {0} such that hx, xip,q < 0 for all x ∈ Ce. We denote by Ce0 ⊂ Ce CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 59

∂Hp,q−1 γm ·y y C

am γ ·a z m m

γm ·z ∂iC

Figure 6. Illustration for the proof of Lemma 11.3 the preimage of C0. The ideal boundary ∂iC lifts to the intersection ∂fiC ⊂ p,q R r {0} of the closure Ce of Ce with the null cone of h·, ·ip,q minus {0}. For 0 0 any x ∈ Ce and x ∈ ∂fiC, we have hx, x ip,q ≤ 0: indeed, this is easily seen by 0 p,q considering tx+x ∈ R , which for small t > 0 must belong to Ce hence have nonpositive norm; see also [DGK3, Lem. 3.6.(1)]. By Lemma 11.3 we have 0 0 in fact hx, x ip,q < 0 for all x ∈ Ce and x ∈ ∂fiC. In particular, this holds for 0 x ∈ Ce0. Now consider x ∈ Ce0 and x ∈ Ce. Since C0 is the convex hull of ∂iC Pk in C, we may write x = i=1 tixi where x1, . . . , xk ∈ ∂fiC and t1, . . . , tk > 0. 0 0 By Lemma 11.3 we have hxi, x ip,q < 0 for all i, hence hx, x ip,q < 0. This proves that C0 is contained in Ω. The open set Ω is properly convex because C has nonempty interior. Corollary 11.5. In the setting of Lemma 11.4, the group Γ acts cocompactly on some closed properly convex subset C0 of Ω with nonempty interior which p,q−1 0 is contained in C ⊂ H , with ∂iC = ∂iC.

Proof. Since the action of Γ on C0 is cocompact, it is easy to check that for any small enough r > 0 the closed r-neighborhood Cr of C0 in (Ω, dΩ) is p,q−1 0 contained in H (see [DGK3, Lem. 6.3]). The set C := Cr ∩ C is then a closed properly convex subset of Ω with nonempty interior, and Γ acts properly discontinuously and cocompactly on C0. Since C0 is also a closed p,q−1 0 0 p,q−1 subset of H , we have ∂iC = C ∩ ∂H = ∂iCr ∩ ∂iC = ∂iC. 0 If ∂iC does not contain any nontrivial segment, then neither does ∂iC for 0 p,q−1 C as in Lemma 11.5. This proves that if Γ is H -convex cocompact (i.e. satisﬁes (4) of Theorem 1.25, i.e. (2) up to switching p and q), then it satisﬁes condition (v) of Theorem 1.15, as announced. Now, Theorem 1.15 states that (v) is equivalent to strong convex cocompactness: this yields the implication (2) ⇒ (1) of Theorem 1.25. Condi- tion (vi) of Theorem 1.15, equivalent to (v), also says that the group Γ is 60 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

p,q word hyperbolic and that the natural inclusion Γ ,→ PO(p, q) is P1 -Anosov. 0 Since in addition the proximal limit set ΛΓ is the ideal boundary ∂iC , we ﬁnd that ΛΓ is negative. This completes the proof of the implication (4) ⇒ (3) in Theorem 1.25.

11.5. End of the proof of Theorem 1.25. Suppose Γ is irreducible and acts convex cocompactly on some nonempty properly convex open subset Ω p+q p+q of P(R ). Let us prove that Γ is strongly convex cocompact in P(R ). p,q−1 Let ΛΓ be the proximal limit set of Γ in ∂H (Definition 2.2 and Re- p,q−1 mark 11.1); it is a nonempty closed Γ-invariant subset of ∂H which is orb contained in the full orbital limit set ΛΩ (Γ) (see Remark 2.3). By min- cor imality (Lemma 4.1.(2)), the set CΩ (Γ) is the convex hull of ΛΓ in Ω. p,q−1 By [DGK3, Prop. 3.7], the set ΛΓ ⊂ ∂H is nonpositive or nonneg- p,q ative, i.e. it lifts to a cone ΛeΓ of R r {0} on which h·, ·ip,q is everywhere nonpositive or everywhere nonnegative; in the first (resp. second) cor p,q−1 p,q−1 p−1,q p,q−1 case CΩ (Γ) is contained in H ∪ ∂H (resp. S ∪ ∂H ) (see [DGK3, Lem. 3.6.(1)]). By Proposition 4.5 (see also [DGK3, Prop. 3.7]), there is a unique largest Γ-invariant properly convex open domain Ωmax of P(V ) containing Ω, namely the projectivization of the interior of the set of 0 p,q 0 0 x ∈ R such that hx, x ip,q ≤ 0 for all x ∈ ΛeΓ (resp. hx, x ip,q ≥ 0 for cor all x ∈ ΛeΓ). Suppose by contradiction that CΩ (Γ) contains a PET. By cor orb p,q−1 Lemma 4.1.(1), we have ∂iCΩ (Γ) = ΛΩ (Γ) ⊂ ∂H , hence the edges of p,q−1 the PET lie in ∂H . In particular, ha, bip,q = ha, cip,q = 0, and so the ⊥ PET is entirely contained in a ∩ Ωmax ⊂ ∂Ωmax, contradicting the fact that cor cor CΩ (Γ) ⊂ Ω ⊂ Ωmax. This shows that CΩ (Γ) does not contain any PET, p+q and so Γ is strongly convex cocompact in P(R ) by Theorem 1.15. 11.6. Proof of Theorem 1.29. The implications (4) ⇒ (3) ⇒ (2) of The- orem 1.29 hold trivially. We now prove (1) ⇔ (2) ⇒ (4). We start with the following observation. ∗ Lemma 11.6. For p, q ∈ N , let Γ be an infinite discrete subgroup of PO(p, q) acting properly discontinuously and cocompactly on a closed convex p,q−1 subset C of H with nonempty interior. Then C has bisaturated boundary p,q−1 if and only if ∂nC does not contain any infinite geodesic line of H . p,q−1 Proof. Suppose ∂nC contains an infinite geodesic line of H . Since C is p,q−1 p,q−1 properly convex and closed in H , this line must meet ∂H in a point of ∂iC, and so C does not have bisaturated boundary. Conversely, suppose C does not have bisaturated boundary. Since ∂iC = p,q−1 p,q Fr(C)∩∂H is closed in P(R ), there exists a ray [y, z) ⊂ ∂nC terminating at a point z ∈ ∂iC. Let (am)m∈N be a sequence of points of [y, z) converging to z (see Figure 6). Since Γ acts cocompactly on C, for any m there exists γm ∈ Γ such that γm · am belongs to a fixed compact subset of ∂nC. Up to taking a subsequence, the sequences (γm · am)m and (γm · y)m and (γm · z)m converge respectively to some points a∞, y∞, z∞ in P(V ). We have a∞ ∈ ∂nC CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 61 and y∞ ∈ ∂iC (because the action of Γ on C is properly discontinuous) and z∞ ∈ ∂iC (because ∂iC is closed). Moreover, a∞ ∈ (y∞, z∞). Thus (y∞, z∞) p,q−1 is an infinite geodesic line of H contained in ∂nC.

p,q−1 Suppose condition (1) of Theorem 1.29 holds, i.e. Γ < PO(p, q) is H - convex cocompact. By Corollary 11.5, the group Γ preserves a properly p,q convex open subset Ω of P(R ) and acts cocompactly on a closed convex 0 p,q−1 subset C of Ω with nonempty interior which is contained in H , and 0 whose ideal boundary ∂iC does not contain any nontrivial segment. As in the proof of Corollary 11.5, for small enough r > 0 the closed r-neighborhood 0 0 p,q−1 Cr of C in (Ω, dΩ) is contained in H . By Lemma 4.1.(1)–(2), we have 0 0 orb 0 ∂iCr = ∂iC = ΛΩ (Γ), hence Cr has bisaturated boundary by Lemma 4.1.(3). 0 p,q−1 In particular, ∂nCr does not contain any inﬁnite geodesic line of H by 0 Lemma 11.6. Thus condition (2) of Theorem 1.29 holds, with Cbisat = Cr. Conversely, suppose condition (2) of Theorem 1.29 holds, i.e. Γ acts properly discontinuously and cocompactly on a closed convex subset Cbisat of p,q−1 p,q−1 H such that ∂nCbisat does not contain any inﬁnite geodesic line of H . By Lemma 11.6, the set Cbisat has bisaturated boundary. By Corollary 4.4, orb the group Γ acts convex cocompactly on Ω := Int(Cbisat) and ΛΩ (Γ) = ∂iCbisat. By Lemma 9.2, the group Γ acts cocompactly on a closed convex cor subset Csmooth ⊃ CΩ (Γ) of Ω whose nonideal boundary is strictly convex 1 orb p,q−1 and C , and ∂iCsmooth = ΛΩ (Γ) ⊂ ∂H (see Lemma 4.1.(1)). In par- p,q−1 ticular, Csmooth is closed in H and condition (4) of Theorem 1.29 holds. Since Cbisat has bisaturated boundary, any inextendable segment in ∂iCbisat is inextendable in ∂Ω = ∂iCbisat ∪∂nCbisat. By Lemma 6.2, if ∂iCsmooth = ∂iCbisat contained a nontrivial segment, then Csmooth would contain a PET, but that p,q−1 is not possible since closed subsets of H do not contain PETs; see Re- mark 11.7 below. Therefore ∂iCsmooth = ∂iCbisat does not contain any non- p,q−1 trivial segment, and so Γ is H -convex cocompact, i.e. condition (1) of Theorem 1.29 holds. This concludes the proof of Theorem 1.29. In the proof we have used the following elementary observation.

∗ p,q−1 Remark 11.7. For p, q ∈ N with p+q ≥ 3, a closed subset of H cannot p,q p,q−1 contain a PET. Indeed, any triangle of P(R ) whose edges lie in ∂H p,q−1 must have interior in ∂H , because in this case the symmetric bilinear form is zero on the projective span.

p,q−1 11.7. H -convex cocompact groups whose boundary is a (p − 1)- p,q−1 sphere. In what follows we use the term standard (p − 1)-sphere in ∂H p,q−1 to mean the intersection with ∂H of the projectivization of a (p + 1)- p+q dimensional subspace of R of signature (p, 1).

Lemma 11.8. For p, q ≥ 1, let Γ be an inﬁnite discrete subgroup of PO(p, q) p,q−1 which is H -convex cocompact, acting properly discontinuously and co- p,q−1 compactly on some closed convex subset C of H with nonempty interior 62 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

p,q−1 whose ideal boundary ∂iC ⊂ ∂H does not contain any nontrivial projective line segment. Let ∂∞Γ be the Gromov boundary of Γ, and let S be any p,q−1 standard (p − 1)-sphere in ∂H . Then p,q−1 (1) the Γ-equivariant boundary map ξ : ∂∞Γ → ∂H , with image ∂iC, p,q−1 is homotopic to an embedding ∂∞Γ → ∂H whose image is contained in S; in particular, ∂∞Γ has Lebesgue covering dimension ≤ p−1. Suppose ∂∞Γ is homeomorphic to a (p − 1)-dimensional sphere and p ≥ q. Then p,q (2) the unique maximal Γ-invariant convex open subset Ωmax of P(R ) p,q−1 containing C (see Proposition 4.5) is contained in H ; (3) any supporting hyperplane of C at a point of ∂nC is the projectivization p,q of a linear hyperplane of R of signature (p, q − 1).

In particular, if q = 2 and ∂∞Γ is homeomorphic to a (p − 1)-dimensional sphere with p ≥ 2, then any hyperplane tangent to ∂nC is spacelike. Lemma 11.8.(1) has the following consequences. Corollary 11.9. Let p, q ≥ 1. p,q−1 (1) Any infinite discrete subgroup of PO(p, q) which is H -convex cocompact has virtual cohomological dimension ≤ p. (2) Let Γ be a word hyperbolic group with connected Gromov boundary p,q ∂∞Γ and virtual cohomological dimension > q. If there is a P1 - Anosov representation ρ :Γ → PO(p, q), then p > q and the group p,q−1 ρ(Γ) is H -convex cocompact. Proof of Corollary 11.9. (1) For a word hyperbolic group Γ, the Lebesgue covering dimension of ∂∞Γ is equal to the virtual cohomological dimension of Γ minus one [BeM]. We conclude using Lemma 11.8.(1). p,q−1 (2) By Corollary 1.27, since ∂∞Γ is connected, the group ρ(Γ) is H - q,p−1 convex cocompact or H -convex cocompact. We conclude using (1). Proof of Lemma 11.8. We work in an affine chart A containing C, and choose p,q coordinates (x1, . . . , xp+q) on R so that the quadratic form h·, ·ip,q takes 2 2 2 2 the usual form x1 + ··· xp − xp+1 − · · · − xp+q, the affine chart A is defined p,q−1 by xp+q 6= 0, and our chosen standard (p − 1)-sphere in ∂H is p,q 2 2 2 S = [x1 : ... : xp : 0 : ... : 0 : xp+q] ∈ P(R ) | x1 + ··· + xp = xp+q . For any 0 ≤ t ≤ 1, consider the map ft : A → A sending [x1 : ... : xp+q] to h √ √ q i x1 : ... : xp : 1 − t xp+1 : ··· : 1 − t xp+q−1 : 1 + t α(xp+1,...,xp+q) xp+q , 2 2 2 where α(xp+1,...,xp+q) = (xp+1 + ··· + xp+q−1)/xp+q. Then (ft)0≤t≤1 restricts p,q−1 p,q−1 to a homotopy of maps A ∩ ∂H → A ∩ ∂H from the identity map p,1 to the map f1|A∩∂Hp,q−1 , which has image S ⊂ P(R ). Thus (ft ◦ ξ)0≤t≤1 p,q−1 defines a homotopy between ξ : ∂∞Γ → ∂H and the continuous map p,q−1 f1 ◦ ξ : ∂∞Γ → ∂H whose image lies in S. By the Cauchy–Schwarz p,q−1 inequality for Euclidean inner products, two points of A ∩ ∂H which CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 63 have the same image under f1 are connected by a line segment in A whose p,q−1 interior lies outside of Hp,q−1. Since C is contained in H , we deduce that the restriction of f1 to ∂iC is injective. Since ∂∞Γ is compact, we obtain that f1 ◦ ξ : ∂∞Γ → S is an embedding. This proves (1). Henceforth we assume that ∂∞Γ is homeomorphic to a (p−1)-dimensional p,q−1 sphere and that p ≥ q. Let ΛΓ = ξ(∂∞Γ) ⊂ ∂H be the proximal limit p+q set of Γ in P(R ). The map f1 :ΛΓ → S is an embedding, hence a homeomorphism since ΛΓ is compact. Let us prove (2). By definition of Ωmax (see Proposition 4.5), it is sufficient p,q−1 ⊥ to prove that any point of ∂H is contained in z for some z ∈ ΛΓ. Therefore it is sufficient to prove that ΛΓ intersects P(W ) for every maximal p,q totally isotropic subspace W of R . Let W be such a subspace. We work in the coordinates from above. Up to changing the first p coordinates by applying an element of O(p), we may assume that in the splitting p,q p−q,0 q,0 0,q R = R ⊕ R ⊕ R 0 0 0 q defined by these coordinates, the space W is {(0, x , −x ) | x ∈ R }. Consider p−1 q−1 0 p,0 p−q,0 q,0 the map ϕ : S → S sending a unit vector (x, x ) ∈ R = R ⊕R to 0 −1 0 ϕ(x, x ) = πq ◦ (f1|ΛΓ ) (x, x , a), 0,q p,q 0,q where a = (0,..., 0, 1) ∈ R and πq : R → R is the projection onto the last q coordinates. Then ϕ is homotopic to a constant map (namely the −1 map sending all points to a) via the homotopy ϕt = πq ◦ ft ◦ (f1|ΛΓ ) . Now q−1 q−1 q,0 consider the restriction ψ : S → S of ϕ to the unit sphere in R , which is also homotopic to a constant map via the restriction of the homotopy ϕt. 0 0 0 If we had ΛΓ ∩ P(W ) = ∅, then we would have ψ(x ) = ϕ(0, x ) 6= −x for all 0 q−1 q x ∈ S ⊂ R , and (1 − t) ψ(x0) + tx0 ψ (x0) = t k(1 − t) ψ(x0) + tx0k q−1 would define a homotopy from ψ to the identity map on S , showing that q−1 a constant map is homotopic to the identity map: contradiction since S p,q−1 is not contractible. This completes the proof of (2): namely, Ωmax ⊂ H . p,q−1 We note that ∂Ωmax ∩ ∂H = ΛΓ, since we saw that f1 maps ∂Ωmax ∩ p,q−1 ∂H injectively to S. Finally, we prove (3). The dual convex Int(C)∗ to Int(C) naturally iden- p+q tifies, via h·, ·ip,q, with a properly convex subset of P(R ) which must be p,q−1 ⊥ contained in Ωmax ⊂ H . By Lemma 11.3, we have C ∩ z = ∅ for any ⊥ z ∈ ∂iC = ΛΓ, hence C ⊂ Ωmax. Hence, a projective hyperplane y supporting C at some point of ∂nC is dual to a point y ∈ Ωmax r ΛΓ, which is p,q−1 contained in H by the previous paragraph. Hence hy, yip,q < 0 and so ⊥ y is the projectivization of a linear hyperplane of signature (p, q − 1).

p,1 11.8. H -convex cocompactness and global hyperbolicity. Recall that a Lorentzian manifold M is said to be globally hyperbolic if it is causal (i.e. 64 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL contains no timelike loop) and for any two points x, x0 ∈ M, the intersection J +(x)∩J −(x0) is compact (possibly empty). Here we denote by J +(x) (resp. J −(x)) the set of points of M which are seen from x by a future-pointing (resp. past-pointing) timelike or lightlike geodesic. Equivalently [CGe], the Lorentzian manifold M admits a Cauchy hypersurface, i.e. an achronal subset which intersects every inextendible timelike curve in exactly one point. We make the following observation; it extends [BM, § 4.2], which focused on the case that Γ ⊂ PO(p, 2) is isomorphic to a uniform lattice of PO(p, 1). The case considered here is a bit more general, see [LM]. We argue in an elementary way, without using the notion of CT-regularity. Proposition 11.10. For p ≥ 2, let Γ be a torsion-free inﬁnite discrete sub- p,1 group of PO(p, 2) which is H -convex cocompact and whose Gromov boundary ∂∞Γ is homeomorphic to a (p−1)-dimensional sphere. For any nonempty p,1 properly convex open subset Ω of H on which Γ acts convex cocompactly, the quotient M = Γ\Ω is a globally hyperbolic Lorentzian manifold. Proof. Consider two points x, x0 ∈ M = Γ\Ω. There exists R > 0 such that any lifts y, y0 ∈ Ω of x, x0 belong to the uniform R-neighborhood C cor + − of CΩ (Γ) in Ω for the Hilbert metric dΩ. Let J (y) (resp. J (y)) be the set of points of Ω which are seen from y by a future-pointing (resp. past- pointing) timelike or lightlike geodesic. By Lemma 11.8.(3), all supporting hyperplanes of C at points of ∂nC are spacelike. We may decompose ∂nC + into two disjoint open subsets, namely the subset ∂n C of points for which the outward pointing normal to a supporting plane is future pointing, and − + the subset ∂n C of points for which it is past pointing. Indeed, ∂n C and − ∂n C are the two path connected components of the complement ∂nC of the embedded (p − 1)-sphere ΛΓ in the p-sphere Fr(C). The set Ω r C similarly + has two components, a component to the future of ∂n C and a component to − + the past of ∂n C. Any point of J (y)∩(ΩrC) lies in the future component of − 0 Ω r C and similarly, any point of J (y ) ∩ (Ω r C) lies in the past component of Ω r C. By proper discontinuity of the Γ-action on C, in order to check that J +(x) ∩ J −(x0) is compact in M, it is therefore enough to check that J +(y) ∩ C and J −(y0) ∩ C are compact in Ω. This follows from the fact that orb the ideal boundary of C is ΛΩ (Γ) (Lemma 4.1.(1)) and that any point of Ω orb sees any point of ΛΩ (Γ) in a spacelike direction (Lemma 11.3).

p,q−1 11.9. Examples of H -convex cocompact groups.

p,q−1 11.9.1. Quasi-Fuchsian H -convex cocompact groups. Let H be a real semisimple Lie group of real rank 1 and τ : H → PO(p, q) a representa- p,q−1 tion whose image contains an element which is proximal in ∂H (see Remark 11.1). By [DGK3, Prop. 7.1], for any word hyperbolic group Γ and any (classical) convex cocompact representation σ0 :Γ → H, p,q (1) the composition ρ0 := τ ◦ σ0 :Γ → PO(p, q) is P1 -Anosov and the p,q−1 proximal limit set Λρ0(Γ) ⊂ ∂H is negative or positive; CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 65

p,q (2) the connected component Tρ0 of ρ0 in the space of P1 -Anosov representations from Γ to PO(p, q) is a neighborhood of ρ0 in Hom(Γ, PO(p, q)) consisting entirely of representations ρ with negative proximal limit p,q−1 set Λρ(Γ) ⊂ ∂H or entirely of representations ρ with positive p,q−1 proximal limit set Λρ(Γ) ⊂ ∂H . Here is an immediate consequence of this result and of Theorem 1.25. p,q−1 Corollary 11.11. In this setting, either ρ(Γ) is H -convex cocompact for q,p−1 all ρ ∈ Tρ0 , or ρ(Γ) is H -convex cocompact (after identifying PO(p, q) with PO(q, p)) for all ρ ∈ Tρ0 . This improves [DGK3, Cor. 7.2], which assumed ρ to be irreducible. Here are two examples. We refer to [DGK3, § 7] for more details. Example 11.12. Let Γ be the fundamental group of a convex cocompact (e.g. closed) hyperbolic manifold M of dimension m ≥ 2, with holonomy m σ0 :Γ → PO(m, 1) = Isom(H ). The representation σ0 lifts to a repre- ∗ sentation σe0 :Γ → H := O(m, 1). For p, q ∈ N with p ≥ m, q ≥ 1, let m,1 p,q τ : O(m, 1) → PO(p, q) be induced by the natural embedding R ,→ R . p,q Then ρ0 := τ ◦ σe0 :Γ → PO(p, q) is P1 -Anosov, and one checks that p,q−1 the set Λρ0(Γ) ⊂ ∂H is negative. Let Tρ0 be the connected component p,q of ρ0 in the space of P1 -Anosov representations from Γ to PO(p, q). By p,q−1 Corollary 11.11, for any ρ ∈ Tρ0 , the group ρ(Γ) is H -convex cocompact. By [Me, Ba], when p = m and q = 2 and when the hyperbolic m-manifold

M is closed, the space Tρ0 of Example 11.12 is a full connected component of Hom(Γ, PO(p, 2)), consisting of so-called AdS quasi-Fuchsian representa- p,1 tions; that ρ(Γ) is H -convex cocompact in that case follows from [Me, BM]. Example 11.13. For n ≥ 2, let 2 n (11.1) τn : SL(R ) −→ SL(R ) 2 be the irreducible n-dimensional linear representation of SL(R ) obtained 2 st n−1 2 from the action of SL(R ) on the (n − 1) symmetric power Sym (R ) ' n R . The image of τn preserves the nondegenerate bilinear form Bn := ⊗(n−1) 2 −ω induced from the area form ω of R . This form is symmetric if n is odd, and antisymmetric (i.e. symplectic) if n is even. Suppose n = 2m + 1 is odd. The symmetric bilinear form Bn has signature (m + 1, m) if m is odd, (11.2) (k , ` ) := n n (m, m + 1) if m is even.

If we identify the orthogonal group O(Bn) (containing the image of τn) with 2 O(kn, `n), then there is a unique τn-equivariant embedding ιn : ∂∞H ,→ kn,`n−1 2 ∂PH , and an easy computation shows that its image Λn := ι(∂∞H ) is negative. For p ≥ kn and q ≥ `n, the representation τn : SL2(R) → O(Bn) ' kn,`n p,q O(kn, `n) and the natural embedding R ,→ R induce a representation τ : H = SL2(R) → PO(p, q) whose image contains an element which is prox- p,q−1 2 kn,`n−1 imal in ∂H , and a τ-equivariant embedding ι : ∂∞H ,→ ∂PH ,→ 66 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

p,q−1 2 p,q−1 ∂PH . The set Λ := ι(∂∞H ) ⊂ ∂PH is negative by construction. Let Γ be the fundamental group of a convex cocompact orientable hyperbolic surface, with holonomy σ0 :Γ → PSL2(R). The representation σ0 lifts to a representation σe0 :Γ → H := SL2(R). Let ρ0 := τ ◦ σe0 :Γ → G := PO(p, q). The proximal limit set Λρ0(Γ) = ι(Λσ0(Γ)) ⊂ Λ is negative. Thus Corol- p,q−1 lary 11.11 implies that ρ0 is H -convex cocompact.

It follows from [L, FG] (see e.g. [BIW2, § 6.1]) that when (p, q) = (kn, `n) or (m + 1, m + 1) and when Γ is a closed surface group, the space Tρ0 of Example 11.13 is a full connected component of Hom(Γ, PO(p, q)), consisting of so-called Hitchin representations. Example 11.13 thus states the following. Corollary 11.14. Let Γ be the fundamental group of a closed orientable hyperbolic surface and let m ≥ 1. For any Hitchin representation ρ :Γ → PO(m + 1, m), the group ρ(Γ) is m+1,m−1 m,m H -convex cocompact if m is odd, and H -convex cocompact if m is even. For any Hitchin representation ρ :Γ → PO(m + 1, m + 1), the group ρ(Γ) m+1,m is H -convex cocompact. By [BIW1, BIW3], when p = m + 1 = 2 and Γ is a closed surface group, the space Tρ0 of Example 11.13 is a full connected component of Hom(Γ, PO(2, q)), consisting of so-called maximal representations. Exam- ple 11.13 thus states the following. Corollary 11.15. Let Γ be the fundamental group of a closed orientable hyperbolic surface and let q ≥ 1. Any connected component of Hom(Γ, PO(2, q)) consisting of maximal representations and containing a Fuchsian represen- 2,q−1 tation ρ0 :Γ → PO(2, 1)0 ,→ PO(2, q) consists entirely of H -convex cocompact representations. 11.9.2. Groups with connected boundary. We now brieﬂy discuss a class of examples that does not necessarily come from the deformation of “Fuchsian” representations as above. Suppose the word hyperbolic group Γ has connected boundary ∂∞Γ (for instance Γ is the fundamental group of a closed negatively-curved Riemann- ian manifold). By [DGK3, Prop. 1.10 & Prop. 3.5], any connected compo- p,q nent in the space of P1 -Anosov representations from Γ to PO(p, q) consists entirely of representations ρ with negative proximal limit set Λρ(Γ) ⊂ p,q−1 ∂H or entirely of representations with ρ with positive proximal limit p,q−1 set Λρ(Γ) ⊂ ∂H . Theorem 1.25 then implies that for any connected component T in the space of P1-Anosov representations of Γ with values in p+q p,q−1 PO(p, q) ⊂ PGL(R ), either ρ(Γ) is H -convex cocompact for all ρ ∈ T , q,p−1 or ρ(Γ) is H -convex cocompact for all ρ ∈ T , as in Corollary 1.27. This applies for instance to the case that Γ is the fundamental group of a closed hyperbolic surface and T is a connected component of Hom(Γ, PO(2, q)) consisting of maximal representations [BIW1, BIW3]. By [GW2], for q = 3 CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 67 there exist such connected components T that consist entirely of Zariski- dense representations, hence that do not come from the deformation of “Fuch- sian” representations as in Corollary 11.15.

12. Examples of groups which are convex cocompact in P(V ) In Section 11.9 we constructed examples of discrete subgroups of PO(p, q) ⊂ p,q p,q−1 PGL(R ) which are H -convex cocompact; these groups are strongly p,q convex cocompact in P(R ) by Theorem 1.25. We now discuss several constructions of discrete subgroups of PGL(V ) which are convex cocompact in P(V ) but which do not necessarily preserve a nonzero quadratic form on V . Based on Theorem 1.15.(D)–(F), another fruitful source of groups that are convex cocompact in P(V ) is the continuous deformation of “Fuchsian” groups, coming from an algebraic embedding. These “Fuchsian” groups can be for instance: (1) convex cocompact subgroups (in the classical sense) of appropriate rank-one Lie subgroups H of PGL(V ), as in Section 11.9.1; (2) discrete subgroups of PGL(V ) dividing a properly convex open subset of some projective subspace P(V0) of P(V ) and acting trivially on a complementary subspace. The groups of (1) are always word hyperbolic; in Section 12.1, we explain how to choose H so that they are strongly convex cocompact in P(V ), and we prove Proposition 1.7. The groups of (2) are not necessarily word hyperbolic; they are convex cocompact in P(V ) by Theorem 1.15.(F), but not necessarily strongly convex cocompact; we discuss them in Section 12.2.2. We also mention other constructions of convex cocompact groups in Sec- tions 12.2.3 and 12.3, which do not involve any deformation.

12.1. “Quasi-Fuchsian” strongly convex cocompact groups. We start by discussing a similar construction to Section 11.9.1, but for discrete subgroups of PGL(V ) which do not necessarily preserve a nonzero quadratic form on V . Let Γ be a word hyperbolic group, H a real semisimple Lie group of real rank one, and τ : H → PGL(V ) a representation whose image contains a proximal element. Any (classical) convex cocompact representation σ0 :Γ → H is Anosov, hence the composition τ ◦ σ0 :Γ → PGL(V ) is P1-Anosov (see [L, Prop. 3.1] and [GW3, Prop. 4.7]). In particular, by The- orem 1.4, the group τ ◦ σ0(Γ) is strongly convex cocompact in P(V ) as soon as it preserves a nonempty properly convex open subset of P(V ). Suppose this is the case. By Theorem 1.17 (see Remark 1.18), the group τ ◦ σ(Γ) remains strongly convex cocompact in P(V ) for any σ ∈ Hom(Γ,H) close enough to σ0. Sometimes τ ◦ σ(Γ) also remains strongly convex cocompact for some σ ∈ Hom(Γ,H) which are continuous deformations of σ0 quite far away from σ0; we now discuss this in view of proving Proposition 1.7. 68 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

12.1.1. Connected open sets of strongly convex cocompact representations. We prove the following. Proposition 12.1. Let Γ be a word hyperbolic group and A a connected open subset of Hom(Γ, PGL(V )) consisting entirely of P1-Anosov representations. (1) Suppose every representation in A is irreducible. If ρ(Γ) is convex cocompact in P(V ) for some ρ ∈ A, then ρ(Γ) is convex cocompact in P(V ) for all ρ ∈ A. (2) Suppose A contains representations ρ which are not irreducible, and that ρ(Γ) is convex cocompact in P(V ) for all these ρ. Then ρ(Γ) is convex cocompact in P(V ) for all ρ ∈ A. The proof of Proposition 12.1 reduces to the following lemma.

Lemma 12.2. Let Γ be a discrete group and (ρm) ∈ Hom(Γ, PGL(V ))N a sequence of discrete and faithful representations converging to a discrete and faithful representation ρ∞ ∈ Hom(Γ, PGL(V )) which is irreducible. If each ρm(Γ) preserves a nonempty properly convex domain Ωm ⊂ P(V ), then so does ρ∞(Γ). Proof of Lemma 12.2. Up to passing to a subsequence, the closed convex subsets Ωm converge to a closed subset C in the Hausdorff topology on compact subsets. The set C is invariant under ρ∞(Γ). Suppose by contradiction that ρ∞(Γ) fails to preserve any nonempty properly convex open domain. Then either C has empty interior or C is not contained in any affine chart. If C has empty interior, then it is contained in a smallest plane of positive codimension and that plane is preserved by ρ∞(Γ), contradicting the fact that 0∗ ρ∞ is irreducible. If C is not contained in any affine chart, then the limit C ∗ of the closures of the dual convex domains Ωm has empty interior. Hence ∗ the dual action of Γ on P(V ) preserves a plane of positive codimension in ∗ P(V ), and so Γ preserves some nontrivial subspace in P(V ), contradicting again the fact that ρ∞ is irreducible. Proof of Proposition 12.1. First, observe that the finite normal subgroups Ker(ρ) ⊂ Γ are constant over ρ ∈ A, since representations of a finite group are rigid up to conjugation. Hence by passing to the quotient group Γ/Ker(ρ), we may assume each representation ρ ∈ A is faithful. Suppose ρ(Γ) is convex cocompact in P(V ) for all ρ ∈ A which are not irreducible. By Theorem 1.17.(D), the property of being convex cocompact in P(V ) is open in Hom(Γ, PGL(V )). By Lemma 12.2, the property of preserving a nonempty properly convex open subset of P(V ) is closed in A. Since A consists entirely of P1-Anosov representations, Theorem 1.15 implies that the property of being convex cocompact in P(V ) is closed in A. 12.1.2. Hitchin representations. We now prove Proposition 1.7 by applying Proposition 12.1.(1) in the following specific context. Let Γ be the fundamental group of a closed orientable hyperbolic surface S. 2 n For n ≥ 2, let τn : SL(R ) → SL(R ) be the irreducible n-dimensional CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 69 linear representation from (11.1). We still denote by τn the representa- 2 n tion PSL(R ) → PSL(R ) obtained by modding out by {±I}. A repre- n sentation ρ ∈ Hom(Γ, PSL(R )) is said to be Fuchsian if it is of the form ρ = τn ◦ρ0 where ρ0 :Γ → PSL(2, R) is discrete and faithful. By definition, a Hitchin representation is a continuous deformation of a Fuchsian representation; the Hitchin component Hitn(S) is the space of Hitchin representations n n ρ ∈ Hom(Γ, PSL(R )) modulo conjugation by PGL(R ). Hitchin [H] used Higgs bundles techniques to parametrize Hitn(S), showing in particular that it is homeomorphic to a ball of dimension (n2−1)(2g−2). Labourie [L] proved that any Hitchin representation is discrete and faithful, and moreover that it is an Anosov representation with respect to a minimal parabolic subgroup n of PSL(R ). In particular, any Hitchin representation is P1-Anosov. In order to prove Proposition 1.7, we first consider the case of Fuchsian representations. 2 n Lemma 12.3. Let ρ :Γ → PSL(R ) → PSL(R ) be Fuchsian. n (1) If n is odd, then ρ(Γ) is strongly convex cocompact in P(R ). (2) If n is even, then the boundary map of the P1-Anosov representation ρ n defines a nontrivial loop in P(R ) and ρ(Γ) does not preserve any n nonempty properly convex open subset of P(R ). For even n, the fact that a Fuchsian representation ρ cannot preserve a n properly convex open subset of P(R ) also follows from [B2, Th. 1.5], since in this case ρ takes values in the projective symplectic group PSp(n/2, R). Proof. (1) Suppose n = 2k + 1 is odd. Then ρ takes values in the projective orthogonal group PSO(k, k + 1) ' SO(k, k + 1), and so ρ(Γ) is strongly n convex cocompact in P(R ) by [DGK3, Prop. 1.17 & 1.19]. (2) Suppose n = 2k is even. Then ρ takes values in the symplectic group Sp(n, R). It is well known that, in natural coordinates identifying ∂∞Γ ' 1 2 n S with P(R ), the boundary map ξ : ∂∞Γ → P(R ) of the P1-Anosov representation ρ is the Veronese curve 2 n−1 n−2 n−2 n−1 n P(R ) 3 [x, y] 7−→ [x , x y, . . . , xy , y ] ∈ P(R ). This map is homotopic to the map [x, y] 7→ [xn−1, 0,..., 0, yn−1] which, since 2 n−1 is odd, is a homeomorphism from P(R ) to the projective plane spanned n by the first and last coordinate vectors, hence is nontrivial in π1(P(R )). n Hence the image of the boundary map ξ crosses every hyperplane in P(R ). It follows that ρ(Γ) does not preserve any properly convex open subset of n P(R ), because the image of ξ must lie in the boundary of any ρ(Γ)-invariant such set. Proof of Proposition 1.7. (1) Suppose n is odd. By [L, FG], all Hitchin repre- n sentations ρ :Γ → PSL(R ) are P1-Anosov. Moreover, these representations ρ are irreducible (see [L, Lem. 10.1]), and ρ(Γ) is convex cocompact in P(V ) as soon as ρ is Fuchsian, by Lemma 12.3. Applying Proposition 12.1.(1), we n obtain that for any Hitchin representation ρ :Γ → PSL(R ) the group ρ(Γ) 70 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL is convex cocompact in P(V ), hence strongly convex cocompact since Γ is word hyperbolic (Theorem 1.15). n (2) Suppose n is even. By Lemma 12.3, for any Fuchsian ρ :Γ → PSL(R ), the image of the boundary map of the P1-Anosov representation ρ defines n a nontrivial loop in P(R ). Since the assignment of a boundary map to an Anosov representation is continuous [GW3, Th. 5.13], we obtain that for any n Hitchin ρ :Γ → PSL(R ), the image of the boundary map of the P1-Anosov n representation ρ defines a nontrivial loop in P(R ). In particular, ρ(Γ) does n not preserve any nonempty properly convex open subset of P(R ), by the same reasoning as in the proof of Lemma 12.3. More generally, ρ does not n preserve any nonempty properly convex subset of P(R ), since any such set would have nonempty interior by irreducibility of ρ.

12.2. Convex cocompact groups coming from divisible convex sets. Recall that any discrete subgroup Γ of PGL(V ) dividing a properly convex open subset Ω ⊂ P(V ) is convex cocompact in P(V ) (Example 1.13). The group Γ is strongly convex cocompact if and only if it is word hyperbolic (Theorem 1.15), and this is equivalent to the fact that Ω is strictly convex (by [B3] or Theorem 1.15 again). After brieﬂy reviewing some examples of nonhyperbolic groups dividing properly convex open sets (Section 12.2.1), we explain how they can be used to construct more examples of convex cocompact groups which are not strongly convex cocompact, either by deformation (Section 12.2.2) or by taking subgroups (Section 12.2.3).

12.2.1. Examples of nonhyperbolic groups dividing properly convex open sets. Examples of virtually abelian, nonhyperbolic groups dividing a properly convex open subset Ω ⊂ P(V ) are easily constructed: see Examples 3.7 and 3.8. In these examples the properly convex (but not strictly convex) set Ω is decomposable: there is a direct sum decomposition V = V1 ⊕ V2 and properly convex open cones Ωe1 of V1 and Ωe2 of V2 such that Ω ⊂ P(V ) is the projection of Ωe1 + Ωe2 := {v1 + v2 | vi ∈ Ωei}. Examples of nonhyperbolic groups dividing an indecomposable properly convex open set Ω can classically be built by letting a uniform lattice of m m SL(R ) act on the Riemannian symmetric space of SL(R ), realized as a m(m−1)/2 properly convex subset of P(R ) (see [B7, § 2.4]). A similar construction works over the complex numbers, the quaternions, or the octonions. Such examples are called symmetric. Benoist [B6, Prop. 4.2] gave further examples by considering nonhyperbolic Coxeter groups

mi,j W = s1, . . . , s5 | (sisj) = 1 ∀ 1 ≤ i, j ≤ 5 CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 71 for which the matrix (mi,j)1≤i,j≤5 is given by 1 3 3 2 2  3 1 3 2 2    3 3 1 m 2    2 2 m 1 ∞ 2 2 2 ∞ 1 with m ∈ {3, 4, 5}. He constructed explicit injective and discrete represen- 4 tations ρ : W → PGL(R ) for which the group ρ(W ) divides a properly 4 n convex open set Ω ⊂ P(R ). Similar examples can be constructed in P(R ) for n = 5, 6, 7 [B6, Rem. 4.3]. These groups admit abelian subgroups isomor- n−2 phic to Z . Ballas–Danciger–Lee [BDL] gave more examples of nonhyperbolic groups 4 dividing properly convex open sets Ω ⊂ P(R ). In particular, if a finite- volume hyperbolic 3-manifold M with torus cusps is infinitesimally projectively rigid relative to the cusps (see [BDL, Def. 3.1]), then the cusp group becomes diagonalizable in small deformations of the holonomy of M and the double DM of the manifold along its torus boundary components admits a properly convex projective structure. The universal cover DMg identifies with 4 a properly convex open subset of P(R ) which is divided by the fundamental group π1(DM). Recently, for n = 5, 6, 7, Choi–Lee–Marquis [CLM] have constructed other examples of injective and discrete representations ρ of certain nonhyperbolic n Coxeter groups W into PGL(R ) for which the group ρ(W ) divides a prop- n erly convex open set Ω ⊂ P(R ). These groups admit abelian subgroups n−3 n−2 isomorphic to Z but not to Z . 12.2.2. Convex cocompact deformations of groups dividing a convex set. Let Γ be a discrete subgroup of PGL(V ) dividing a nonempty properly convex open subset Ω of P(V ). By Theorem 1.17.(D)–(F), any small deformation of the inclusion of Γ into a larger PGL(V ⊕ V 0) is convex cocompact in 0 P(V ⊕ V ). 4 For nonhyperbolic Γ, this works well in particular for V = R . Indeed, Benoist [B6] described a beautiful structure theory for indecomposable di- 4 visible convex sets in P(R ): the failure of strict convexity is due to the presence of PETs; the stabilizer in Γ of each PET of Ω is virtually the fun- 2 damental group Z of a torus, and the group Γ splits as an amalgamated product or HNN extension over the stabilizer. Thus the inclusion of (the lift ± 4 4 n0 to SL (R ) of) Γ into a larger PGL(R ⊕ R ) may be deformed nontrivially by a Johnson–Millson bending construction [JM] along PETs. Such small deformations give examples of nonhyperbolic subgroups which are convex 4 n0 cocompact in P(R ⊕ R ), but do not divide a properly convex set. The deformations may be chosen irreducible (indeed this is generically the case). 12.2.3. Convex cocompact groups coming from divisible convex sets by taking subgroups. We now explain that nonhyperbolic groups dividing a properly 72 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

4 convex open set Ω ⊂ P(R ) admit nonhyperbolic subgroups that are still 4 convex cocompact in P(R ), but that do not divide any properly convex 4 open subset of P(R ). 4 Let Γ be a torsion-free, nonhyperbolic, discrete subgroup of PGL(R ) 4 dividing a nonempty properly convex open subset Ω of P(R ), as above. By [B6], the PETs in Ω descend to a finite collection of pairwise disjoint planar tori and Klein bottles in the closed manifold N := Γ\Ω. Cutting N along this collection yields a collection of compact manifolds M with torus and Klein bottle boundary components. Each such M is a compact convex projective manifold with boundary, but from topological considerations, Benoist [B6] proved that M r ∂M also admits a finite-volume hyperbolic structure (in which the boundary tori and Klein bottles are cusps). The universal cover of M identifies with a convex subset CM of Ω whose interior is a connected component of the complement in Ω of the union of all PETs, and whose nonideal boundary ∂nCM is a disjoint union of PETs. The fundamental group of M identifies with the subgroup ΓM of Γ that preserves CM , and it acts properly discontinuously and cocompactly on CM .

Lemma 12.4. The action of ΓM on Ω is convex cocompact. cor Proof. By Lemma 4.1.(1), it is enough to check that CΩ (ΓM ) ⊂ CM . Since each component of ∂nCM is planar, CM is the convex hull of its ideal boundary orb ∂iCM , and so it is enough to show that ΛΩ (ΓM ) ⊂ ∂iCM . Suppose by contradiction that this is not the case: namely, there exists y ∈ Ω r CM and a sequence (γm) in ΓM such that (γm · y) converges to some y∞ ∈ orb ΛΩ (ΓM ) r ∂iCM . Let z be the point of ∂nCM which is closest to y for the Hilbert metric dΩ; it is contained in a PET of ∂nCM . Let a, b ∈ ∂Ω be such that a, y, z, b are aligned in that order. Up to taking a subsequence, we may assume that (γm · z), (γm · a), and (γm · b) converge respectively to some z∞, a∞, b∞ ∈ ∂Ω, with z∞ ∈ ∂iCM and a∞, y∞, z, b∞ aligned in that order. The cross-ratio [a∞, y∞, z∞, b∞] is equal to [a, y, z, b] ∈ (1, +∞), hence the points a∞, y∞, z, b∞ are pairwise distinct and contained in a segment of ∂Ω. However, any segment on ∂Ω lies on the boundary of some PET [B6]. Thus we have found a PET whose closure intersects the closure of CM but is not contained in it; its closure must cross the closure of a second PET on ∂nCM , contradicting the fact [B6] that PETs have disjoint closures. The deformation construction of Ballas–Danciger–Lee [BDL] discussed in section §12.2.1 yields many more examples, similar to Lemma 12.4, of con- 4 vex cocompact subgroups in PGL(R ) for which the nonideal boundary of the convex core consists of a union of PETs. As in §12.2.1, let M be a compact, connected, orientable 3-manifold with a union of tori as boundary whose interior admits a finite-volume complete hyperbolic structure which is infinitesimally projectively rigid relative to the cusps in the sense of [BDL, Def. 3.1]. Then [BDL, Th. 1.3] gives properly convex projective deformations of the hyperbolic structure on M where each boundary component is CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 73 a totally geodesic (i.e. planar) torus. It is an easy corollary of techniques 4 in [BDL] that the holonomy representations in PGL(R ) for such convex 4 projective structures have image that is convex cocompact in P(R ). One way to see this is that each planar torus boundary component admits an explicit strictly convex thickening. 12.3. Convex cocompact groups as free products. Being convex cocompact in P(V ) is a much more flexible property than dividing a properly convex open subset of P(V ), and there is a rich world of examples, which we shall explore in forthcoming work [DGK5]. In particular, we shall prove the following.

Proposition 12.5 ([DGK5]). Let Γ1 and Γ2 be infinite discrete subgroups of PGL(V ) which are convex cocompact in P(V ) but do not divide any nonempty properly convex open subset of P(V ). Then there exists g ∈ PGL(V ) such −1 that the group generated by Γ1 and gΓ2g is isomorphic to the free product Γ1 ∗ Γ2 and is convex cocompact in P(V ). This yields many examples of nonhyperbolic convex cocompact groups, for instance Coxeter groups by taking e.g. Γ1 = Γ2 to be the subgroup of n n+1 PGL(R ) of Example 3.8, embedded into PGL(V ) for V = R . Appendix A. Some open questions Here we list some open questions about discrete subgroups Γ of PGL(V ) that are convex cocompact in P(V ). The case that Γ is word hyperbolic boils down to Anosov representations by Theorem 1.4 and is reasonably under- stood; on the other hand, the case that Γ is not word hyperbolic corresponds to a new class of discrete groups whose study is still in its infancy. Most of the following questions are interesting even in the case that Γ divides a (non- strictly convex) properly convex open subset of P(V ). We fix a discrete subgroup Γ of PGL(V ) acting convex cocompactly on a nonempty properly convex open subset Ω of P(V ). Question A.1. Which finitely generated subgroups Γ0 of Γ are convex cocompact in P(V )? Such subgroups are necessarily quasi-isometrically embedded in Γ, by Corollary 10.2. Conversely, when Γ is word hyperbolic, any quasi-isometrically embedded (or equivalently quasi-convex) subgroup Γ0 of Γ is convex cocompact in P(V ): indeed, the boundary maps for the P1-Anosov representation 0 Γ ,→ PGL(V ) induce boundary maps for Γ ,→ PGL(V ) that make it P1- 0 Anosov, hence Γ is convex cocompact in P(V ) by Theorem 1.4. When Γ is not word hyperbolic, Question A.1 becomes more subtle. For 2 3 instance, let Γ ' Z be the subgroup of PGL(R ) consisting of all diagonal matrices whose entries are powers of some fixed t > 0; it is convex cocompact 3 0 0 in P(R ) (see Example 3.7). Any cyclic subgroup Γ = hγ i of Γ is quasi- 0 isometrically embedded in Γ. However, Γ is convex cocompact in P(V ) if and only if γ0 has distinct eigenvalues (see Examples 3.9.(1)–(2)). 74 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL

Question A.2. Assume Ω is indecomposable. Under what conditions is Γ relatively hyperbolic, relative to a family of virtually abelian subgroups? This is always the case if dim(V ) ≤ 3. If dim(V ) = 4 and Γ divides Ω, then this is also seen to be true from work of Benoist [B6]: in this case there are finitely many conjugacy classes of PET stabilizers in Γ, which are virtually 2 Z , and Γ is relatively hyperbolic with respect to the PET stabilizers (using [Dah, Th. 0.1]). However, when dim(V ) = m(m − 1)/2 for some m ≥ 3, we can take for Ω ⊂ P(V ) the projective realization of the Riemannian symmetric space of SLm(R) (see [B7, § 2.4]) and for Γ a uniform lattice of SLm(R): this Γ is not relatively hyperbolic with respect to any subgroups, see [BDM]. Question A.3. If Γ is not word hyperbolic, must there be a properly em- k bedded maximal k-simplex invariant under some subgroup isomorphic to Z for some k ≥ 2? This question is a specialization, to the class of convex-cocompact subgroups of PGL(V ), of the following more general question (see [Be, Q 1.1]): if G is a finitely generated group admitting a finite K(G, 1), must it be word hyperbolic as soon as it does not contain a Baumslag–Solitar group BS(m, n)? 2 Here we view Z as the Baumslag–Solitar group BS(1, 1). Note that our convex cocompact group Γ ⊂ PGL(V ) cannot contain BS(m, n) for m, n ≥ 2 since it does not contain any unipotent element by Theorem 1.17.(C). In light of the equivalence (ii) ⇔ (vi) of Theorem 1.15, we ask the following: Question A.4. When Γ is not word hyperbolic, is there a dynamical description, similar to P1-Anosov, that characterizes the action of Γ on P(V ), orb for example in terms of ΛΩ (Γ) and divergence of Cartan projections? While this question is vague, a good answer could lead to the definition of new classes of nonhyperbolic discrete subgroups in other higher-rank reductive Lie groups for which there is not necessarily a good notion of convexity. A variant of the following question was asked by Olivier Guichard. Question A.5. In Hom(Γ, PGL(V )), does the interior of the set of naively convex cocompact representations consist of convex cocompact representations? Here we say that a representation is naively convex cocompact (resp. convex cocompact) if it is faithful and if its image is naively convex cocompact in P(V ) (resp. convex cocompact in P(V )) in the sense of Definition 1.9 (resp. Definition 1.11).

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Department of Mathematics, The University of Texas at Austin, 1 Univer- sity Station C1200, Austin, TX 78712, USA E-mail address: [email protected]

CNRS and Université Lille 1, Laboratoire Paul Painlevé, 59655 Villeneuve d’Ascq Cedex, France E-mail address: [email protected]

CNRS and Institut des Hautes Études Scientifiques, Laboratoire Alexan- der Grothendieck, 35 route de Chartres, 91440 Bures-sur-Yvette, France E-mail address: [email protected]